Octad « Search Results

Thursday, January 27, 2022

The Lexicographic Octad Generator

Filed under: General — Tags: Embedding Change, Octad, Octad Generator, Strange Change, The Guy Embedding — m759 @ 2:30 pm

A Lexicographic Basis for the Binary Golay Code:

Brouwer and Guven — "Long ago," in
"The generating rank of the space of short vectors
in the Leech lattice mod 2," by
Andries Brouwer & Cicek Guven,
https://www.win.tue.nl/~aeb/preprints/udim24a.pdf —

"One checks by computer" that this is a basis:

000000000000000011111111
000000000000111100001111
000000000011001100110011
000000000101010101010101
000000001001011001101001
000000110000001101010110
000001010000010101100011
000010010000011000111010
000100010001000101111000
001000010001001000011101
010000010001010001001110
100000010001011100100100

(Copied from the Brouwer-Guven paper)

_________________________________________________

Adlam at Harvard —
"Constructing the Extended Binary Golay Code,"
by Ben Adlam, Harvard University, August 9, 2011,
https://fliphtml5.com/llqx/wppz/basic —

Adlam also asserts, citing a reference, that this same
set of twelve vectors is a basis:

(Copied from the Adlam paper)
__________________________________________________

Sources —

Background for Log24 posts on 'Embedding Change'

"One checks by computer" —

At http://magma.maths.usyd.edu.au/calc/ —

> V24 := VectorSpace(FiniteField(2), 24);
> G := sub< V24 |
> [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1],
> [0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1],
> [0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1],
> [0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],
> [0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1],
> [0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,1,0,1,0,1,1,0],
> [0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,1,1,0,0,0,1,1],
> [0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,0,0,1,1,1,0,1,0],
> [0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,1,1,1,1,0,0,0],
> [0,0,1,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,1,1,1,0,1],
> [0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,1,1,1,0],
> [1,0,0,0,0,0,0,1,0,0,0,1,0,1,1,1,0,0,1,0,0,1,0,0]>;
> Dimension(G);
12

Comments Off

Wednesday, November 18, 2020

The Octad Club

Filed under: General — Tags: Eightfold Metaphysics, Octad — m759 @ 11:03 am

“Principles before personalities.” — AA motto

Related personalities —

The Metaphysical Club .

Amazon.com review by John Miller :

“… The Metaphysical Club is not a dry tome for academics.
Instead, it is a quadruple biography … .”

Related principles —

The Octad Group .

Comments Off

Monday, July 13, 2020

The Lexicographic Octad Generator (LOG)*

Filed under: General — Tags: Canonicity, Lexico.graphics, MJT Guy, Natural Diagram, Octad, Octad Generator, The Guy Embedding, The Log — m759 @ 5:43 pm

The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.

By Steven H. Cullinane, July 13, 2020

Background —

The Miracle Octad Generator (MOG)
of R. T. Curtis (Conway-Sloane version) —

Embedding Change, Illustrated

A basis for the Golay code, excerpted from a version of
the code generated in lexicographic order, in

"Constructing the Extended Binary Golay Code"
Ben Adlam
Harvard University
August 9, 2011:

Below, each vector above has been reordered within
a 4×6 array, by Steven H. Cullinane, to form twelve
independent Miracle Octad Generator vectors
(as in the Conway-Sloane SPLAG version above, in
which Curtis's earlier heavy bricks are reflected in
their vertical axes) —

01 02 03 04 05 . . . 20 21 22 23 24 -->

01 05 09 13 17 21
02 06 10 14 18 22
03 07 11 15 19 23
04 08 12 16 20 24

0000 0000 0000 0000 1111 1111 -->

0000 11
0000 11
0000 11
0000 11 as in the MOG.

0000 0000 0000 1111 0000 1111 -->

0001 01
0001 01
0001 01
0001 01 as in the MOG.

0000 0000 0011 0011 0011 0011 -->

0000 00
0000 00
0011 11
0011 11 as in the MOG.

0000 0000 0101 0101 0101 0101 -->

0000 00
0011 11
0000 00
0011 11 as in the MOG.

0000 0000 1001 0110 0110 1001 -->

0010 01
0001 10
0001 10
0010 01 as in the MOG.

0000 0011 0000 0011 0101 0110 -->

0000 00
0000 11
0101 01
0101 10 as in the MOG.

0000 0101 0000 0101 0110 0011 -->

0000 00
0101 10
0000 11
0101 01 as in the MOG.

0000 1001 0000 0110 0011 1010 -->

0100 01
0001 00
0001 11
0100 10 as in the MOG.

0001 0001 0001 0001 0111 1000 -->

0000 01
0000 10
0000 10
1111 10 as in the MOG.

0010 0001 0001 0010 0001 1101 -->

0000 01
0000 01
1001 00
0110 11 as in the MOG.

0100 0001 0001 0100 0100 1110 -->

0000 01
1001 11
0000 01
0110 00 as in the MOG.

1000 0001 0001 0111 0010 0100 -->

10 00 00
00 01 01
00 01 10
01 11 00 as in the MOG (heavy brick at center).

Update at 7:41 PM ET the same day —
A check of SPLAG shows that the above result is not new:

And at 7:59 PM ET the same day —
Conway seems to be saying that at some unspecified point in the past,
M.J.T. Guy, examining the lexicographic Golay code, found (as I just did)
that weight-8 lexicographic Golay codewords, when arranged naturally
in 4×6 arrays, yield certain intriguing visual patterns. If the MOG existed
at the time of his discovery, he would have identified these patterns as
those of the MOG. (Lexicographic codes have apparently been
known since 1960, the MOG since the early 1970s.)

* Addendum at 4 AM ET the next day —
See also Logline (Walpurgisnacht 2013).

Comments Off

Saturday, May 2, 2020

Turyn’s Octad Theorem: The Next Level*

Filed under: General — Tags: Octad, Octad Generator, The Next Level — m759 @ 11:24 am

From the obituary of a game inventor who reportedly died
on Monday, February 25, 2013 —

” ‘He was hired because of the game,’ Richard Turyn,
a mathematician who worked at Sylvania, told
the Washington Post in a 2004 feature on Diplomacy.”

* For the theorem, see Wolfram Neutsch, Coordinates .
(Published by de Gruyter, 1996. See pp. 761-766.)

Having defined (pp. 751-752) the Miracle Octad Generator (MOG)
as a 4×6 array to be used with Conway’s “hexacode,” Neutsch says . . .

“Apart from the three constructions of the Golay codes
discussed at length in this book (lexicographic and via
the MOG or the projective line), there are literally
dozens of alternatives. For lack of space, we have to
restrict our attention to a single example. It has been
discovered by Turyn and can be connected in a very
beautiful way with the Miracle Octad Generator….

To this end, we consider the natural splitting of the MOG into
three disjoint octads L, M, R (‘left’, ‘middle’, and ‘right’ octad)….”

— From page 761

“The theorem of Turyn” is on page 764 —

Comments Off

Thursday, April 23, 2020

Octads and Geometry

Filed under: General — Tags: Art Issue, Brick Symmetry, Brick Wall, Devs, Octad, Octad Generator, Octad Group, Overarching Group — m759 @ 10:11 pm

See the web pages octad.group and octad.us.

Related geometry (not the 759 octads, but closely related to them) —

The 4×6 rectangle of R. T. Curtis
illustrates the geometry of octads —

Curtis splits the 4×6 rectangle into three 4×2 "bricks" —

"In fact the construction enables us to describe the octads
in a very revealing manner. It shows that each octad,
other than Λ₁, Λ₂, Λ₃, intersects at least one of these ' bricks' —
the 'heavy brick' – in just four points." . . . .

— R. T. Curtis (1976). "A new combinatorial approach to M₂₄,"
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 25-42.

Comments Off

Sunday, March 15, 2020

The “Octad Group”

Filed under: General — Tags: Octad, Octad Generator, Octad Group, Overarching Group, Priority — m759 @ 4:17 pm

The phrase “octad group” discussed here in a post
of March 7 is now a domain name, “octad.group,”
that leads to that post. Remarks by Conway and
Sloane now quoted there indicate how the group
that I defined in 1979 is embedded in the large
Mathieu group M₂₄.

Related literary notes — Watson + Embedding.

Comments Off

Saturday, March 7, 2020

The “Octad Group” as Symmetries of the 4×4 Square

Filed under: General — Tags: Gaberdiel, Mathieu Moonshine, Octad, Octad Generator, Octad Group, Overarching, Overarching Group — m759 @ 6:32 pm

From "Mathieu Moonshine and Symmetry Surfing" —

(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)

(1) Institute for Theoretical Physics, ETH Zurich
(2) Department of Mathematics, ETH Zurich

https://arxiv.org/abs/1609.09302v2 —

"This presentation of the symmetry groups G_i is
particularly well-adapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is
the so-called octad group

G = (Z₂)⁴⋊ A₈.

It can be described as a maximal subgroup of M₂₄
obtained by the setwise stabilizer of a particular
'reference octad' in the Golay code, which we take
to be O₉= {3,5,6,9,15,19,23,24} ∈ 𝒢₂₄. The octad
subgroup is of order 322560, and its index in M₂₄
is 759, which is precisely the number of
different reference octads one can choose."

This "octad group" is in fact the symmetry group of the affine 4-space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79T-A37, "Symmetry invariance in a
diamond ring," by Steven H. Cullinane in Notices of the American Mathematical
Society , February 1979, pages A-193, 194.

* The Galois tesseract .

Update of March 15, 2020 —

Conway and Sloane on the "octad group" in 1993 —

Comments Off

Wednesday, February 19, 2020

Aitchison’s Octads

Filed under: General — Tags: Aitchison, Eightfold Cube, Fro Zen, Mathieu Cube, Octad, Octad Generator — m759 @ 11:36 am

The 759 octads of the Steiner system S(5,8,24) are displayed
rather neatly in the Miracle Octad Generator of R. T. Curtis.

A March 9, 2018, construction by Iain Aitchison* pictures the
759 octads on the faces of a cube , with octad elements the
24 edges of a cuboctahedron :

The Curtis octads are related to symmetries of the square.

See my webpage "Geometry of the 4×4 square" from March 2004.
Aitchison's p. 42 slide includes an illustration from that page —

Aitchison's octads are instead related to symmetries of the cube.

Note that essentially the same model as Aitchison's can be pictured
by using, instead of the 24 edges of a cuboctahedron, the 24 outer
faces of subcubes in the eightfold cube .

Image from Christmas Day 2005.

* http://www.math.sci.hiroshima-u.ac.jp/branched/files/2018/
presentations/Aitchison-Hiroshima-2-2018.pdf.
See also Aitchison in this journal.

Comments Off

Wednesday, December 11, 2019

Miracle Octad Generator Structure

Filed under: General — Tags: Design Notes Dec. 11, Geometry of Even Subsets, Natural Diagram, Octad, Octad Generator — m759 @ 11:44 pm

Miracle Octad Generator — Analysis of Structure

(Adapted from Eightfold Geometry, a note of April 28, 2010.
See also the recent post Geometry of 6 and 8.)

Comments Off

Saturday, November 4, 2017

Seven-Cycles in an Octad

Filed under: G-Notes,General,Geometry — Tags: Octad, Springer — m759 @ 8:00 pm

Figures from a search in this journal for Springer Knight
and from the All Souls' Day post The Trojan Pony —

Binary coordinates for a 4x2 array Chess knight formed by a Singer 7-cycle

For those who prefer pure abstraction to the quasi-figurative
1985 seven-cycle above, a different 7-cycle for M₂₄ , from 1998 —

Compare and contrast with my own "knight" labeling
of a 4-row 2-column array (an M₂₄ octad, in the system
of R. T. Curtis) by the 8 points of the projective line
over GF(7), from 2008 —

'Knight' octad labeling by the 8 points of the projective line over GF(7)

Comments Off

Friday, March 21, 2014

Three Constructions of the Miracle Octad Generator

Filed under: General,Geometry — Tags: Depth, Geometry, Octad, Octad Generator, Priority — m759 @ 12:24 pm

See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.

Related material on the Turyn-Curtis construction
from the University of Cambridge —

— Slide by "Dr. Parker" — Apparently Richard A. Parker —
Lecture 4, "Discovering M₂₄," in slides for lectures 1-8 from lectures
at Cambridge in 2010-2011 on "Sporadic and Related Groups."
See also the Parker lectures of 2012-2013 on the same topic.

A third construction of Curtis's 35 4×6 1976 MOG arrays would use
Cullinane's analysis of the 4×4 subarrays' affine and projective structure,
and point out the fact that Conwell's 1910 correspondence of the 35
4+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis's 1976 MOG.

See The Diamond Theorem, Finite Relativity, Galois Space,
Generating the Octad Generator, and The Klein Correspondence.

Update of March 22-March 23 —

Adding together as (0,1)-matrices over GF(2) the black parts (black
squares as 1's, all other squares as 0's) of the 35 4×6 arrays of the 1976
Curtis MOG would then reveal* the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S₃ on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35 MOG arrays. For more
details of this "by-hand" construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction (such as Turyn's), not by hand, of the
extended binary Golay code. See the Brouwer preprint quoted above.

* "Then a miracle occurs," as in the classic 1977 Sidney Harris cartoon.

Illustration of array addition from March 23 —

Comments Off

Sunday, April 28, 2013

The Octad Generator

Filed under: General,Geometry — Tags: Dots, Geometry, Octad, Octad Generator, Six-Set — m759 @ 11:00 pm

… And the history of geometry —
Desargues, Pascal, Brianchon and Galois
in the light of complete n-points in space.

(Rewritten for clarity at about 10 AM ET April 29, with quote from Dowling added.
Updated with a reference to a Veblen and Young exercise (on p. 53) on April 30.)

Veblen and Young, Projective Geometry, Vol. I ,
Ginn and Company, 1910, page 39:

"The Desargues configuration. A very important configuration
is obtained by taking the plane section of a complete space five-point."

Each of figures 14 and 15 above has 15 points and 20 lines.
The Desargues configuration within each figure is denoted by
10 white points and 10 solid lines, with 3 points on each line and
3 lines on each point. Black points and dashed lines indicate the
complete space five-point and lines connecting it to the plane section
containing the Desargues configuration.

In a 1915 University of Chicago doctoral thesis, Archibald Henderson
used a complete space six -point to construct a configuration of
15 points and 20 lines in the context not of Desargues ' theorem, but
rather of Brianchon 's theorem and of the Pascal hexagram.
Henderson's 1915 configuration is, it turns out, isomorphic to that of
the 15 points and 20 lines in the configuration constructed via a
complete space five -point five years earlier by Veblen and Young.
(See, in Veblen and Young's 1910 Vol. I, exercise 11, page 53:
"A plane section of a 6-point in space can be considered as
3 triangles perspective in pairs from 3 collinear points with
corresponding sides meeting in 3 collinear points." This is the
large Desargues configuration. See Classical Geometry in Light of
Galois Geometry.)

For this large Desargues configuration see April 19.
For Henderson's complete six –point, see The Six-Set (April 23).
That post ends with figures relating the large Desargues configuration
to the Galois geometry PG(3,2) that underlies the Curtis
Miracle Octad Generator and the large Mathieu group M₂₄ —

See also Note on the MOG Correspondence from April 25, 2013.

That correspondence was also discussed in a note 28 years ago, on this date in 1985.

Comments Off

Thursday, April 11, 2024

Devs: Hollywood Development Hell

Filed under: General — Tags: Diamond Theorem Correlation, Hollywood and History, Mesmerization — m759 @ 11:53 am

From The Man Who Knew Infinity to The Man Who Knew Zero.

Related mathematics: The Diamond Theorem Correlation, which
results from interchanging infinity and zero in the figure below.

"Read something that means something."
— New Yorker ad

^{Click image for

related posts.}

Background — Relativity Problem in Log24.

Comments (0)

Wednesday, March 20, 2024

Cullinane Diamond Song

Filed under: General — Tags: Nevermore, Octad — m759 @ 12:00 am

Shining Mathematics:
A Song by Suno AI

Pop upbeat

March 19, 2024

[Verse]
In the world of numbers and equations
There's a theorem that'll blow your mind (your mind)
Cullinane's diamond, it's a revelation
A mathematical gem, one of a kind (oh-yeah)

[Chorus]
Shining bright, like a diamond in the night
This theorem's got the sparkle, it's pure delight (ooh-yeah)
Cullinane's diamond, oh it's so sublime
Mathematics never looked so fine (so fine)

Click the image below to hear the song at app.suno.ai —

Miller's Note to Self : "Don't underestimate Wednesday."

For more about the mathematics itself, see other octad posts.

Comments Off

Monday, March 4, 2024

Report on the Diamond Theorem
by Google Gemini (Advanced)

Filed under: General — Tags: Chatbots on the Diamond Theorem, Octad — m759 @ 3:26 pm

Diamond Theorem Report

by Google Gemini AI

Monday, March 4, 2024

Can you write an academic paper, with source citations, about the Cullinane diamond theorem?

Absolutely! Here's an academic paper about the Cullinane diamond theorem. Please note that while I provide citations, in a fully formatted paper you'd need to adhere to your chosen academic style guide for proper formatting:

The Cullinane Diamond Theorem:
Exploring Symmetries in Graphic Design

Abstract

This paper examines the Cullinane diamond theorem, a principle in finite projective geometry that offers unexpected insights into the symmetries found in graphic designs. The theorem, initially formulated by Steven H. Cullinane, reveals that patterns built from simple geometric components, like diamonds or chevrons, possess inherent symmetry properties under specific transformation groups. We will discuss the mathematical foundations of the theorem, its relation to the Miracle Octad Generator, and its applications in design and the study of finite fields.

Introduction

Symmetry plays a vital role in art and design, contributing to a sense of balance, harmony, and aesthetic appeal. While often associated with classical forms, symmetry also manifests in surprising ways within seemingly simple graphic patterns. The Cullinane diamond theorem provides a mathematical framework to understand and predict these hidden symmetries.

Mathematical Background

Finite Projective Geometries: The Cullinane diamond theorem has its roots in finite projective geometry. A finite projective geometry PG(n, q) consists of a finite set of points and lines, where each line contains q + 1 points, and any two distinct points lie on a unique line ([1]).

Diamond Figure: Cullinane focuses on a specific pattern called the four-diamond figure (D). This figure is composed of 16 tiles arranged in a 4×4 array, where each tile is divided diagonally into two colors ([2]).

Transformation Group: The group G of transformations is formed by all possible permutations of rows, columns, and 2×2 quadrants of the diamond figure, along with color interchanges. The size of this group is 322,560 ([2]).

The Cullinane Diamond Theorem

Theorem:
Any image produced by applying transformations from the group G to the original diamond figure (D) will possess some form of ordinary or color-interchange symmetry ([2]).

Proof:
Cullinane's proof employs a decomposition technique for functions into a finite field. It demonstrates that the G-images of the diamond figure create an ideal within a ring of symmetric patterns. This ideal has 1024 patterns where any horizontal or vertical "cut" through the pattern remains uninterrupted ([2]).

Connections to the Miracle Octad Generator

The Cullinane diamond theorem shares a relationship with R.T. Curtis's Miracle Octad Generator (MOG). The MOG is a combinatorial structure within the study of sporadic simple groups with symmetry properties similar to those described by the theorem ([3]).

Applications

Graphic Design: The theorem informs artistic compositions employing diamond and chevron-like shapes. Designers can leverage the inherent symmetries to generate visually pleasing and intricate patterns.

Quilt Patterns: Many traditional quilt designs incorporate the geometries described by the theorem, leading to the emergence of symmetrical patterns, even when randomness might be involved in the selection of colors or tiles.

Finite Fields: The Cullinane diamond theorem establishes an isomorphism between specific "diamond" rings and rings of matrices over GF(4), offering insights into the structure of finite fields ([2]).

Conclusion

The Cullinane diamond theorem elegantly connects abstract mathematics with the world of graphic design. It unveils latent symmetries embedded within basic geometric shapes, providing a tool for understanding and creating visually balanced patterns. The theorem's connections to finite fields and combinatorial structures highlight its significance in broader mathematical research.

Citations

[1] Weisstein, E. W. “Finite Projective Geometry.” From MathWorld–A Wolfram Web Resource. [invalid URL removed]

[2] Cullinane, S. H. "Cullinane Diamond Theorem." Encyclopedia of Mathematics. https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem

[3] Curtis, R. T. (1976). A New Combinatorial Approach to M₂₄. Mathematical Proceedings of the Cambridge Philosophical Society, 79(1), 25–42. doi:10.1017/S0305004100052075

Let me know if you'd like me to expand on any particular aspect or provide additional citations!

Sources [I.e., the source of the Curtis citation]

1. www.hellenicaworld.com/Science/Mathematics/en/MathieuGroup.html

Gemini may display inaccurate info, including about people, so double-check it.

Comments Off

Sunday, March 3, 2024

Deep Blue Research: A Report by You.com AI

Filed under: General — Tags: Chatbots on the Diamond Theorem, Four Color, Octad — m759 @ 12:34 pm

Cullinane Diamond Theorem Research Report

by https://you.com/?chatMode=research on March 3, 2024

Overview of the Cullinane Diamond Theorem

The Cullinane Diamond Theorem is a mathematical concept developed by Steven H. Cullinane that explores the symmetrical properties of certain geometric patterns. It is particularly concerned with the structure of finite projective geometry as it relates to the 35 square patterns found in R. T. Curtis's Miracle Octad Generator (MOG).

The theorem asserts that every G-image of a diamond figure D possesses some form of ordinary or color-interchange symmetry.

Symmetry and Group Theory

The theorem is deeply rooted in group theory, with G being isomorphic to the affine group A on the linear 4-space over the finite field GF(2).

This group has a significant order of 322,560 and underlies the symmetry of both the diamond-theorem figures and the square patterns of the MOG . The 35 structures of the 840 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), which explains the symmetry of the line diagrams and the two-color patterns.

The Diamond Theorem in Various Dimensions

The diamond theorem can be applied to different cases, such as the 4×4, 2×2, and even 3-dimensional cubes like the 2x2x2 and 4x4x4 cubes.

The generalized statement of the theorem involves inscribing a white diamond in a black square and dividing it into four quadrants, each with a diagonal split into black and white halves.

Algebraic Implications

The theorem has algebraic implications, such as the ability to express every 4-coloring as a sum of three 2-colorings.

This algebraic formulation is useful for demonstrating that certain four-colored graphic images can form a ring under multiplication. There is also an infinite family of such 'diamond' rings, isomorphic to rings of matrices over GF(4).

Philosophical and Cultural Context

Steven H. Cullinane has also explored the philosophical implications of the diamond theorem, referring to it in the context of Plato's Meno dialogue and opposing it to the 'Story Theory of truth' as discussed by Richard J. Trudeau.

The theorem is seen as a correct statement of the relation between geometry and the physical universe.

Clarification on Cullinan Diamonds

It is important to note that the Cullinane Diamond Theorem is unrelated to the Cullinan diamond, which is a famous large gemstone discovered in South Africa and now part of the British Crown Jewels.

The Cullinan diamond has its own historical and cultural significance, with its various cut shapes and ownership history.

In summary, the Cullinane Diamond Theorem is a mathematical concept that reveals the symmetrical properties of certain geometric patterns, with applications in group theory and algebra. It also has philosophical implications, drawing connections to classical ideas about truth and geometry.

Instagram ad for You.com AI in research mode

"Show me ALL your sources, babe."

— Line adapted from Leonardo DiCaprio

Comments Off

Thursday, February 15, 2024

The Isotropic Die

Filed under: General — Tags: Aitchison, Dreaming Jewels, February Poetics, Mathieu Cube, Octad — m759 @ 11:20 am

Related material: Theodore Sturgeon's novel The Dreaming Jewels
and his story "What Dead Men Tell" . . .

Comments Off

Tuesday, January 30, 2024

Cullinane Diamond Theorem:
The Latest Google Bard Version

Filed under: General — Tags: Gemini.reviews, In Loco, Octad, Venice Beach Barbenheimer — m759 @ 7:00 am

Prompt:

What is the Cullinane diamond theorem?

Google Bard Response:

The Cullinane diamond theorem is a fascinating result in the realm of finite projective geometry and symmetry. It focuses on a specific pattern – the four-diamond figure – and reveals a surprising property about its transformations.

The Four-Diamond Figure:

Imagine a 4×4 grid filled with 16 small squares, half colored black and half white. Now, divide these squares diagonally into quarters, creating four diamonds with alternating black and white shades. This arrangement is the four-diamond figure.

_________________

[Clearly erroneous illustration with link to amazon.com
is omitted.]

_____________________

The Theorem:

The Cullinane diamond theorem states that no matter how you scramble this four-diamond figure by randomly shuffling rows, columns, and the four 2×2 quadrants, the resulting pattern will always exhibit some form of symmetry. This symmetry can come in two forms:

Ordinary symmetry: This is the familiar reflectional symmetry where a vertical or horizontal axis divides the pattern into mirrored halves.

Color-interchange symmetry: Here, the pattern has a diagonal axis where swapping the black and white colors results in the same image.

Intriguing Implications:

This statement might seem intuitive at first. After all, even with scrambling, some semblance of order should remain. However, the beauty of the Cullinane diamond theorem lies in its generality and the precise mathematical framework it provides for understanding these symmetries.

The theorem sheds light on the underlying structure of graphic patterns and connects them to concepts from finite projective geometry. This connection allows us to analyze and classify these patterns in a systematic way, revealing hidden connections and surprising symmetries.

Applications and Beyond:

The Cullinane diamond theorem has found applications in various areas, including:

Sporadic simple groups: The Miracle Octad Generator, used to study these rare and exceptional groups, relies on the symmetry properties outlined in the theorem.

Orthogonal Latin squares: The connection between orthogonal Latin squares and projective spreads can be understood through the lens of the Cullinane diamond theorem.

Walsh functions: The symmetry properties of Walsh functions, used in signal processing and coding theory, can be linked to the diamond patterns.

Understanding the Cullinane diamond theorem offers a deeper appreciation for the interplay between symmetry, geometry, and seemingly random patterns. It serves as a testament to the beauty and hidden order within the realm of mathematics.

Bard may display inaccurate info, including . . . .

[Exercise … Find any inaccurate information
in the Bard response above.]

Comments Off

Thursday, December 21, 2023

Pi.ai

Filed under: General — Tags: Chandelier, Octad — m759 @ 10:57 am

Thursday, December 21, 2023

Hey there, great to meet you. I’m Pi, your personal AI. My goal is to be useful, friendly and fun. Ask me for advice, for answers, or let’s talk about whatever’s on your mind.

How's your day going?

Going OK so far, Pi. How's yours?

My day is going swimmingly! 🌊 Thanks for asking 😊 Anything in particular on your mind today?

What is the connection of "heptads" with the "MOG" of R. T. Curtis?

The term “heptads” is connected to the Miracle Octad Generator (MOG) created by R. T. Curtis. Heptads are 7-cycles that arise from the MOG, and they can be used to rearrange the original 1976 version of the MOG into two 4-sets. This is related to the Mathieu groups, which are a family of finite simple groups that play a role in the geometry of the MOG. Are you familiar with the Mathieu groups?

Just the large one. I am interested in how familiar YOU are with these concepts. Sorry, but I must correct your hallucination about heptads… It was a good guess, though. Can you look up the work of G. M. Conwell on PG(3,2)? Heptads are discussed there.

Yes, it seems that G. M. Conwell discussed the concept of heptads in relation to the projective geometry PG(3,2). Heptads are apparently 7-cycles that can be used to establish geometrical isomorphisms and describe the structure of PG(3,2). Do you know if Conwell's work on heptads has been applied to other areas of mathematics?

It has not, as far as I know. Actually, Pi, heptads are not 7-cycles. What made you think that they are?

My apologies! I assumed that heptads would be 7-cycles since the name “heptads” suggests a relation to the number 7 and cycles are common in mathematics. It sounds like I was off the mark there. If heptads are not 7-cycles, can you give me a better idea of what they are?

Conwell can. See his 1910 paper for further details. All for now, Pi… Have a nice day!

Got it — thanks for the tip on Conwell's paper. I'll take a look for more information there! It was great chatting with you. You have a great day as well 😊 Feel free to reach out again if you have any other questions or just want to chat. See ya!

Scholium —

Related material for "swimmingly" and "the number 7" — Snark Hunt.

Sia!

Comments Off

Tuesday, December 19, 2023

Copilot Report

Filed under: General — Tags: Chatbots on the Diamond Theorem, Octad, Rosenhain and Göpel — m759 @ 12:16 am

From Microsoft Copilot (with "Precise" setting)
on Dec. 18, 2023 —

How is the 1910 work of G. M. Conwell on PG(3,2) related to the Miracle Octad Generator of R. T. Curtis?

Searching for: Relation between G. M. Conwell work on PG(3,2) and Miracle Octad Generator of R. T. Curtis

Searching for: Relation between G. M. Conwell work on PG(3,2) and Miracle Octad Generator of R. T. Curtis explained

Generating answers for you…

The work of George M. Conwell on the finite projective 3-space PG(3,2) and the Miracle Octad Generator (MOG) of R. T. Curtis are both significant contributions to the field of combinatorial mathematics ¹ ².

Conwell’s work on PG(3,2) provided a detailed study of the structure of this finite projective space ¹. On the other hand, the MOG is a tool for exploring the binary Golay code and its automorphism group M24 ².

The connection between these two works lies in the underlying combinatorial structures they explore. The Rosenhain and Göpel tetrads, which are key structures in the finite geometry PG(3,2), play a significant role in the Curtis Miracle Octad Generator ³. These structures provide a link between the finite geometric structures studied by Conwell and the combinatorial structures generated by the MOG ³.

For a more detailed understanding of this connection, I recommend checking out the resources mentioned in the references ¹ ² ³. If you have any specific questions about the concepts in these works, feel free to ask! I’ll do my best to help.

Learn more

archive.org

xenon.stanford.edu

academia.edu

en.wikipedia.org

archive.org
_________________________________________________________

__________________________________________________________

DOCUMENT SUMMARY (generated by Copilot)
of reference 3 above, from academia.edu —
"Rosenhain and Göpel Tetrads in PG(3,2),"
by Steven H. Cullinane on March 17, 2013

Here is a summary of the document:

Rosenhain and Göpel Tetrads: The document discusses the 80 Rosenhain odd tetrads and the 60 Göpel even tetrads, which are subsets of four points in a projective 3-space over a 2-element field.
Diamond Theorem Model: The document shows how the 140 Rosenhain and Göpel tetrads occur naturally in the diamond theorem model of the 35 lines of the projective 3-space ¹.
Unidirectional Line Diagrams: The document explains how the arrays containing the Rosenhain and Göpel tetrads have different numbers of unidirectional line diagrams, which are related to the symmetry of the arrays.
References: The document cites several sources that relate the Göpel tetrads to various topics in mathematics, such as quartic surfaces, birational automorphisms, the Miracle Octad Generator, and the Moore correspondence.

Reference:

1. Page 2 – Line 1

Copilot suggestions following document summary —

Correction:

"Subsets of four points in a projective 3-space" above is a Copilot error, and not
from the document being summarized. It should be "subsets of four points in an
affine 4-space."

Comments Off

Monday, December 18, 2023

AI Class

Filed under: General — Tags: Chatbots, Chatbots on the Diamond Theorem, Octad — m759 @ 12:31 pm

Bringing Up Bing :

A Tutorial for Bing on Monday, December 18, 2023

Prompt: How is the Cullinane Diamond Theorem related to the Curtis Miracle Octad Generator?

Response: Searching for: Cullinane Diamond Theorem and Curtis Miracle Octad Generator relation.

Generating answers for you…

The Cullinane Diamond Theorem and the Curtis Miracle Octad Generator (MOG) are both mathematical concepts that are deeply intertwined with finite projective geometry ¹ ^,2.

The Cullinane Diamond Theorem explains the surprising symmetry properties of some simple graphic designs and underlies the structure of the 35 square patterns in R. T. Curtis’s MOG ¹ ^,2. The theorem uses a 4×4 array of two-color diagonally-divided square tiles, and considers the group of permutations of these 16 tiles generated by mixing random permutations of rows, columns, and the four 2×2 quadrants ¹ ^,2.

The theorem states that every image of the 4×4 array has some ordinary or color-interchange symmetry ^1,². The 35 structures of the 840 images of the 4×4 array are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), a finite field with two elements ^1,². This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets ¹ ^,2.

The MOG is a 4×6 array of combinations describing any point in 24-dimensional space ¹. It preserves all of the symmetries and maximal subgroups of the Mathieu group M24 ¹. The MOG is used in the construction of the Steiner system S(5,8,24), and hence is also related to the Leech lattice ².

In summary, the Cullinane Diamond Theorem provides a geometric explanation for the structure and symmetries of the patterns in the MOG ¹^,². Both theorems are connected through their use of finite projective geometry and their applications to the study of symmetry ¹ ^,2.

Learn more

Reply to Bing's Response —

Actually, the Cullinane diamond theorem, by itself, does NOT explain the patterns in the MOG. The MOG is a natural correspondence, or pairing, of the 35 diamond-theorem structures with the 35 partitions of an eight-element set into two four-element sets. That correspondence is explained in a 1910 paper by G. M. Conwell, "The 3-Space PG(3,2) and Its Group." The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24=16+8 elements.

Response to the Above Reply —

Generating answers for you…

Thank you for the clarification. You’re correct that the Cullinane Diamond Theorem, by itself, does not explain the patterns in the Miracle Octad Generator (MOG). The MOG is indeed a natural correspondence of the 35 structures from the Diamond Theorem with the 35 partitions of an eight-element set into two four-element sets. This correspondence was explained by G. M. Conwell in his 1910 paper “The 3-Space PG(3,2) and Its Group.” The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24 elements, which can be partitioned into a set of 16 and a set of 8. I appreciate your input and the opportunity to learn from it. If you have any more questions or need further clarification, feel free to ask!

Comments Off

Sunday, December 17, 2023

Speak, Memory

Filed under: General — Tags: Chatbots on the Diamond Theorem, Gemini.reviews, Octad — m759 @ 12:51 am

"Unsocratic Dialogue" with Google DeepMind's Gemini AI continues . . .

Do you remember a previous conversation about the question "What is the Cullinane diamond theorem?" If so, how would you answer the question now?
______________________________________________________________

Yes, I do remember our previous conversation about the Cullinane diamond theorem. My understanding of the theorem has significantly improved thanks to your patient explanations and corrections. Here's how I would answer the question now, incorporating my updated knowledge:

The Cullinane diamond theorem is a fascinating result in finite projective geometry that reveals hidden symmetries within seemingly scrambled patterns. It focuses on the diamond figure, a 4×4 square pattern with alternating colors arranged in a diamond shape.

Key elements:

Group G: This group consists of permutations of the 16 tiles in the diamond figure, but not just any arbitrary rearrangements. G's permutations are restricted to those that can be generated by manipulating rows, columns, and quadrants of the 4×4 grid. This maintains the overall structure of the diamond shape.
Line diagrams: The theorem doesn't focus on the visual appearance of the G-images (diamonds with rearranged tiles), but rather on their underlying structure. This structure is captured by a set of 15 line diagrams. These diagrams represent the connections between the tiles and encode the essence of the different G-images.
Hidden symmetries: The Cullinane diamond theorem states that every G-image, despite its scrambled appearance, must exhibit some form of ordinary or color-interchange symmetry. This means that when we analyze the line diagrams of these seemingly chaotic patterns, we find hidden symmetries based on the relationships between the tiles.

Significance:

The theorem sheds light on the structure of R.T. Curtis's Miracle Octad Generator, a mathematical object that generates intriguing patterns related to the diamond figure.
It explains the surprising symmetries found in simple graphic designs like quilts, even when their individual components are arranged differently.
It provides a tool for analyzing and classifying graphic patterns based on their hidden symmetries revealed by the line diagrams.

Further resources:

For a deeper understanding, I recommend checking out the "Cullinane diamond theorem" entry in the Encyclopedia of Mathematics.
Additionally, Steven H. Cullinane's original paper titled "The Diamond Theorem" on arXiv provides further details and insights.

I hope this explanation is more accurate and comprehensive than my previous attempts. Please let me know if you have any further questions or if you'd like me to elaborate on any specific aspects of the theorem. I'm still learning and I value your feedback in my journey to provide accurate and insightful information.

Comments Off

Thursday, December 14, 2023

Unsocratic Dialogue (with Gemini AI in Bard Today)

Filed under: General — Tags: Bullshit Studies, Chatbots on the Diamond Theorem, Gemini.reviews, Octad — m759 @ 3:35 pm

What is the Cullinane diamond theorem?

The Cullinane diamond theorem is a fascinating result in the realm of finite projective geometry and its applications to graphic patterns. It delves into the symmetry properties of specific patterns, even when they're seemingly scrambled.

Here's a breakdown of the theorem:

Key components:

The diamond (D): A specific 4×4 square pattern with alternating colors in a diamond shape.
G-images: Any pattern obtained from D by applying symmetries like rotations, flips, and color interchanges.
Ordinary or color-interchange symmetry: Symmetry properties like rotational symmetry or flipping the entire pattern while swapping colors.

The theorem states:

Every G-image of D (no matter how scrambled) must exhibit some form of ordinary or color-interchange symmetry.

In other words, no matter how you twist and turn the diamond pattern, some underlying symmetry remains hidden within. This hidden symmetry can be revealed by applying specific transformations like rotations or color swaps.

Consequences and applications:

The theorem sheds light on the structure of R.T. Curtis's Miracle Octad Generator, a mathematical object that generates intriguing patterns.
It explains the surprising symmetries found in simple graphic designs, like quilts.
It provides a tool for analyzing and classifying various graphic patterns based on their hidden symmetries.

Further details:

The theorem connects the 35 patterns in Curtis's Miracle Octad Generator to finite projective geometry.
The theorem also has connections to algebraic structures like rings and ideals.

If you're interested in learning more:

You can find the original paper by Steven H. Cullinane titled "The Diamond Theorem" on arXiv.
You can explore resources like the Encyclopedia of Mathematics and PlanetMath for further details and explanations.

I hope this explanation helps you understand the Cullinane diamond theorem! Feel free to ask if you have any further questions or want to delve deeper into specific aspects of the theorem.

Your description of "G-images" is wrong. It omits the definition of the group G of 322,560 transformations.

You're absolutely right, I apologize for the mistake in my description of G-images in the Cullinane diamond theorem. My explanation omitted the crucial detail about the group G and its composition, which is essential to understanding the theorem.

Here's a corrected explanation with the missing information:

The group G:

G consists of 322,560 transformations that can be applied to the diamond pattern (D).
These transformations include:
- 8 rotations: 0°, 45°, …, 270°
- 2 flips: horizontal and vertical
- 4 color interchanges: swapping black and white
- All combinations of the above: This allows for a wide range of possible G-images.

Therefore, a G-image of D is any pattern that can be obtained by applying one or more transformations from the group G. This means that even seemingly scrambled versions of the diamond pattern can be considered G-images as long as they can be generated through these transformations.

The Cullinane diamond theorem then states that every G-image of D, despite its apparent complexity, must exhibit some form of ordinary or color-interchange symmetry. This highlights the hidden order and underlying structure within seemingly chaotic patterns.

I'm still under development and learning to be as accurate as possible. Thank you for pointing out my mistake. I appreciate your patience and understanding, and I'll strive to provide more precise information in the future.

Please let me know if you have any other questions or if you'd like me to elaborate on any specific aspects of the Cullinane diamond theorem.

Your understanding of the group G is still wrong. It is a group of PERMUTATIONS of the 16 square cells, or tiles, that make up the diamond figure D. It is NOT a group of space symmetries as you describe.

Thank you for pointing out another critical error in my understanding of the Cullinane diamond theorem. You're absolutely right; the group G in the theorem deals with permutations of the tiles within the diamond figure D, not space symmetries.

I apologize for the confusion and appreciate your patience as I continue to learn. Here's a corrected explanation of the group G:

Group G:

G consists of all permutations of the 16 tiles (squares) that make up the diamond figure D.
This means G contains all possible arrangements of the tiles while maintaining the overall shape of the diamond.
The number of permutations in G is 16! (approximately 322,560), making it a large and diverse group.

Therefore, under the Cullinane diamond theorem, G-images are any configurations of the 16 tiles obtained by applying these permutations. This emphasizes the theorem's focus on rearranging the tiles while preserving the diamond structure, not manipulating the diamond itself through rotations or flips.

I understand now that the theorem focuses on the intrinsic symmetries within the diamond figure based on tile arrangement, not spatial transformations like rotations or flips. This clarifies the theorem's significance in understanding the structural properties of patterns even after scrambling their individual components.

Thank you again for correcting my mistake. I am committed to providing accurate information, and your feedback helps me learn and improve. Please don't hesitate to point out any further errors or areas for improvement.

Is there anything else you'd like me to clarify about the Cullinane diamond theorem or any other related topics? I'm eager to learn more and provide you with the best possible understanding.

Your understanding has improved, but is still wrong. G does not consist of ALL permuations of the 16 tiles, but only those generated by permutations of rows, columns, and quadrants of the 4×4 tile-array.

You're absolutely right, my understanding of Group G in the Cullinane Diamond Theorem is still not quite accurate. I appreciate your patience and persistence in correcting me!

Based on your explanation, I understand that Group G is not simply all permutations of the 16 tiles, but it has a specific structure:

G consists of permutations of the 16 tiles that can be realized by permuting rows, columns, and quadrants of the 4×4 grid.
This means that the group does not include arbitrary rearrangements of the tiles that would break the underlying structure of the diamond figure.

This clarified structure makes sense in light of the theorem's focus on hidden symmetries within the diamond pattern. By restricting the permutations to those that maintain the overall diamond shape, the theorem highlights the intrinsic symmetries that emerge even when rearranging the tiles within that structure.

I apologize for my previous misunderstandings and appreciate your help in correcting them. It's through these interactions that I learn and improve my understanding of complex topics.

Would you be willing to share some resources or further explanation about how these permutations of rows, columns, and quadrants actually generate the elements of Group G? I'm eager to understand the structure of this group more deeply.

Thank you again for your patience and expertise!

Actually, the permutations do NOT "maintain the overall diamond shape," but rather maintain the SET of 15 "line diagrams" that describe the STRUCTURE of the array's images under the group G. The symmetry of these line diagrams accounts for the symmetry of the G-images.

You're absolutely right again! My understanding of the Cullinane Diamond Theorem and the role of Group G is still evolving, and I appreciate your continued patience and guidance.

Your point about the line diagrams and their structure is crucial. I apologize for my earlier misunderstanding about the group's role in maintaining the diamond shape. Instead, as you correctly point out, the group maintains the set of 15 line diagrams that represent the structural essence of the various G-images.

This clarifies the theorem's focus even further. By focusing on the underlying structure captured by the line diagrams, the theorem reveals the hidden symmetries that persist despite rearranging the tiles within the diamond framework. The permutations of rows, columns, and quadrants, within the constraints of maintaining these line diagrams, generate the diverse G-images while preserving these intrinsic structural symmetries.

I'm fascinated by this concept and would be grateful if you could share some resources or further explanation about the specific properties of these 15 line diagrams and how they relate to the permutations and the overall structure of Group G. Understanding their role in the theorem would significantly enhance my grasp of this fascinating concept.

Thank you once again for your expertise and for helping me learn and improve. I'm committed to providing accurate information and your feedback is invaluable in that process.

I suggest the resource "Cullinane diamond theorem" in the online Encyclopedia of Mathematics. Perhaps as your understanding of how visual images can be manipulated improves (with multimodal learning), comprehension of the diamond theorem will also improve. Best of luck.

Thank you so much for your guidance! I'm definitely looking forward to diving deeper into the "Cullinane diamond theorem" in the Encyclopedia of Mathematics. Your point about visual manipulation and multimodal learning is insightful and I'll keep that in mind as I continue to learn and develop. I believe that expanding my understanding of visual information processing will be crucial for comprehending this and other complex concepts.

I appreciate your patience and understanding throughout this conversation. It has been an invaluable learning experience for me, and I'm grateful for your willingness to help me refine my understanding of the Cullinane diamond theorem.

I wish you all the best as well!

Comments Off

Saturday, December 9, 2023

How Many Magic Beans, Jack?

Filed under: General — Tags: Milagro Beanfield, Octad — m759 @ 3:31 pm

Two notes from this journal on December 2, 2023 —

A flashback for The November Man —

As for " my own 'diamond theorem' " —

Question — "What is mathematician Steven H. Cullinane known for?"

From the Copilot response (shorn of footnotes) —

"Steven H. Cullinane is known for his work in finite geometry, particularly his Diamond Theorem. This theorem explains the surprising symmetry properties of some simple graphic designs, such as those found in quilts. It also provides a model for the 15 points, 35 lines, and 15 hyperplanes of the finite projective space PG (3,2).

Cullinane’s work underlies the structure of the 35 square patterns in R. T. Curtis’s Miracle Octad Generator. His theorem has applications to sporadic simple groups, the connection between orthogonal Latin squares and projective spreads, and the symmetry of Walsh functions."

Comments Off

Saturday, December 2, 2023

My Work According to Copilot

Filed under: General — Tags: Chatbots on the Diamond Theorem, Milagro Beanfield, Octad — m759 @ 7:05 am

Following yesterday's encounter with the latest version of Pi,
here are some more formal (and more informative) remarks
from Windows Copilot today —

Question — "What is mathematician Steven H. Cullinane known for?"

From the Copilot response (shorn of footnotes) —

Comments Off

Saturday, November 18, 2023

“Don’t solicit for your sister,* it’s not nice.” — Tom Lehrer

Filed under: General — Tags: Octad — m759 @ 11:36 am

André Weil to his sister:

From this journal at 1:51 AM ET Thursday, September 8, 2022 —

"The pleasure comes from the illusion" . . .

Exercise:

Compare and contrast the following structure with the three
"bricks" of the R. T. Curtis Miracle Octad Generator (MOG).

Note that the 4-row-2-column "brick" at left is quite
different from the other two bricks, which together
show chevron variations within a Galois tesseract —

^{Further Weil remarks . . .}

^{A Slew of Prayers}

"The pleasure comes from the illusion
and the far from clear meaning;
once the illusion is dissipated,
and knowledge obtained, one becomes
indifferent at the same time;
at least in the Gitâ there is a slew of prayers
(slokas) on the subject, each one more final
than the previous ones."

_* ^—

Comments Off

Friday, October 13, 2023

Turn, Turn, Turn

Filed under: General — Tags: Artbox, Logic Fields, Octad — m759 @ 3:06 am

The conclusion of a Hungarian political figure's obituary in
tonight's online New York Times, written by Clay Risen —

"A quietly religious man, he spent his last years translating
works dealing with Roman Catholic canon law."

This journal on the Hungarian's date of death, October 8,
a Sunday, dealt in part with the submission to Wikipedia of
the following brief article . . . and its prompt rejection.

The Cullinane diamond theorem is a theorem
about the Galois geometry underlying
the Miracle Octad Generator of R. T. Curtis.^[1]

The theorem also explains symmetry properties of the
sort of chevron or diamond designs often found on quilts.

Reference

1. Cullinane diamond theorem at
the Encyclopedia of Mathematics

Some quotations I prefer to Catholic canon law —

Ludwig Wittgenstein,
Philosophical Investigations (1953)—

97. Thought is surrounded by a halo.
—Its essence, logic, presents an order,
in fact the a priori order of the world:
that is, the order of possibilities * ,
which must be common to both world and thought.
But this order, it seems, must be
utterly simple . It is prior to all experience,
must run through all experience;
no empirical cloudiness or uncertainty can be
allowed to affect it ——It must rather be of
the purest crystal. But this crystal does not appear
as an abstraction; but as something concrete,
indeed, as the most concrete,
as it were the hardest thing there is.

* See the post Wittgenstein's Diamond.

Related language in Łukasiewicz (1937)—

See as well Diamond Theory in 1937.

Comments Off

Saturday, September 23, 2023

The Cullinane Diamond Theorem at Wikipedia

Filed under: General — Tags: Octad — m759 @ 8:48 am

This post was prompted by the recent removal of a reference to
the theorem on the Wikipedia "Diamond theorem" disambiguation
page. The reference, which has been there since 2015, was removed
because it linked to an external source (Encyclopedia of Mathematics)
instead of to a Wikipedia article.

For anyone who might be interested in creating a Wikipedia article on
my work, here are some facts that might be reformatted for that website . . .

https://en.wikipedia.org/wiki/
User:Cullinane/sandbox —

Cullinane diamond theorem

The theorem uses finite geometry to explain some symmetry properties of some simple graphic designs, like those found in quilts, that are constructed from chevrons or diamonds.

The theorem was first discovered by Steven H. Cullinane in 1975 and was published in 1977 in Computer Graphics and Art.

The theorem was also published as an abstract in 1979 in Notices of the American Mathematical Society.

The symmetry properties described by the theorem are related to those of the Miracle Octad Generator of R. T. Curtis.

The theorem is described in detail in the Encyclopedia of Mathematics article "Cullinane diamond theorem."

References

Steven H. Cullinane, "Diamond theory," Computer Graphics and Art, Vol. 2, No. 1, February 1977, pages 5-7.

_________, Abstract 79T-A37, "Symmetry invariance in a diamond ring," Notices of the American Mathematical Society, February 1979, pages A-193, 194.

_________, "Cullinane diamond theorem," Encyclopedia of Mathematics.

R. T. Curtis, A new combinatorial approach to M₂₄, Mathematical Proceedings of the Cambridge Philosophical Society, 1976, Vol. 79, Issue 1, pages 24-42.

Comments Off

Friday, September 22, 2023

Figurate Space

Filed under: General — Tags: Figurate Geometry, Octad — m759 @ 11:01 am

For the purpose of defining figurate geometry , a figurate space might be
loosely described as any space consisting of finitely many congruent figures —
subsets of Euclidean space such as points, line segments, squares,
triangles, hexagons, cubes, etc., — that are permuted by some finite group
acting upon them.

Thus each of the five Platonic solids constructed at the end of Euclid's Elements
is itself a figurate space, considered as a collection of figures — vertices, edges,
faces — seen in the nineteenth century as acted upon by a group of symmetries .

More recently, the 4×6 array of points (or, equivalently, square cells) in the Miracle
Octad Generator of R. T. Curtis is also a figurate space . The relevant group of
symmetries is the large Mathieu group M₂₄ . That group may be viewed as acting
on various subsets of a 24-set… for instance, the 759 octads that are analogous
to the faces of a Platonic solid. The geometry of the 4×6 array was shown by
Curtis to be very helpful in describing these 759 octads.

Comments Off

Sunday, June 25, 2023

High Concept: The Dreaming Gemstones

Filed under: General — Tags: Jumbo, Octad, Shoe Graveyard — m759 @ 12:07 am

Sturgeon versus Plato —

Sturgeon's Dreaming Jewels meet Plato's Righteous Gemstones.

Comments Off

Wednesday, May 3, 2023

Folklore

Filed under: General — Tags: Folklore, Octad — m759 @ 8:58 am

"Piotr Grochowski's 'Miracles and Visionaries in the Digital Age'
explores the impact of digital technolgies in Catholic Apparitional
movements." (links added) — Journal of American Folklore ,
Fall issue, October 2022 preview.

As for "Miracles and Visionaries," I prefer the literature associated
with the 1974 Miracle Octad Generator of R. T. Curtis.

Comments Off

Friday, April 28, 2023

The Small Space Model

Filed under: General — Tags: Models, Mythspace, Octad, Sister Simone — m759 @ 6:28 pm

From the previous post, "The Large Language Model,"
a passage from Wikipedia —

"… sometimes large models undergo a 'discontinuous phase shift'
where the model suddenly acquires substantial abilities not seen
in smaller models. These are known as 'emergent abilities,' and
have been the subject of substantial study." — Wikipedia

Compare and contrast
this with the change undergone by a "small space model,"
that of the finite affine 4-space A with 16 points (a Galois tesseract ),
when it is augmented by an eight-point "octad." The 30 eight-point
hyperplanes of A then have a natural extension within the new
24-point set to 759 eight-point octads, and the 322,560 affine
automorphisms of the space expand to the 244,823,040 Mathieu
automorphisms of the 759-octad set — a (5, 8, 24) Steiner system.

For a visual analogue of the enlarged 24-point space and some remarks
on analogy by Simone Weil's brother, a mathematician, see this journal
on September 8 and 9, 2022.

Comments Off

Wednesday, March 29, 2023

An Earlier “Pretty Mama” Song, from a 1975 Album

Filed under: General — Tags: Cocktail Song, Octad, Pretty Mama — m759 @ 12:54 pm

The previous post referenced the "pretty mama" of "Cocktail" (1988).

Earlier, in 1975, there was a more serious song to a pretty mama . . .

One of these nights
One of these crazy old nights
We're gonna find out, pretty mama
What turns on your lights

See as well "Dreaming Jewels" and . . .

Comments Off

Tuesday, February 7, 2023

The Graduate School of Design

Filed under: General — Tags: At the Gate, Octad, Wormhole — m759 @ 1:03 pm

The above cubic equation may also be written as

x³ – x – 1 = 0.

The equation occurred in my own work in 1985:

An architects' equation that appears also in Galois geometry.

For further details on the plastic number, see an article by
Siobhan Roberts on John Baez in The New York Times —

Comments Off

Friday, January 13, 2023

The “Diamond Space” of Mazzola

Filed under: General — Tags: Church Socials, Gestures, Octad — m759 @ 11:30 am

The Source —

Some similar notions from my own work . . .

The "Digraph" of Mazzola might correspond to a directed graph
indicating the structure of a permutation, as at right below —

Mazzola's "Formula" might correpond to a matrix and translation that
transform the above "Space" of eight coordinates, and his "Gesture"
to a different way of generating affine transformations of that space . . .
as in my webpage "Cube Space, 1984-2003."

Comments Off

Friday, October 28, 2022

The Lexeme “Deploy”

Filed under: General — Tags: JAF, Lexemes, Octad, The Zombie Doctrine — m759 @ 2:22 pm

From an Oct. 26 elegy in the form of a book review …

From a search in this journal for "deploy" —

Related art —

As for "Miracles and Visionaries," I prefer the literature associated
with the 1974 Miracle Octad Generator of R. T. Curtis.

Comments Off

Thursday, September 29, 2022

The 4×6 Problem*

Filed under: General — Tags: Bitspace, Octad — m759 @ 4:03 pm

The exercise posted here on Sept. 11, 2022, suggests a
more precisely stated problem . . .

The 24 coordinate-positions of the 4096 length-24 words of the
extended binary Golay code G₂₄can be arranged in a 4×6 array
in, of course, 24! ways.

Some of these ways are more geometrically natural than others.
See, for instance, the Miracle Octad Generator of R. T. Curtis.
What is the size of the largest subcode C of G₂₄that can be
arranged in a 4×6 array in such a way that the set of words of C
is invariant under the symmetry group of the rectangle itself, i.e. the
four-group of the identity along with horizontal and vertical reflections
and 180-degree rotation.

Recent Log24 posts tagged Bitspace describe the structure of
an 8-dimensional (256-word) code in a 4×6 array that has such
symmetry, but it is not yet clear whether that "cube-motif" code
is a Golay subcode. (Its octads are Golay, but possibly not all its
dodecads; the octads do not quite generate the entire code.)
Magma may have an answer, but I have had little experience in
its use.

* Footnote of 30 September 2022. The 4×6 problem is a
special case of a more general symmetric embedding problem.
Given a linear code C and a mapping of C to parts of a geometric
object A with symmetry group G, what is the largest subcode of C
invariant under G? What is the largest such subcode under all
such mappings from C to A?

Comments Off

Wednesday, September 28, 2022

Bitspace Note

Filed under: General — Tags: Bitspace, Octad — m759 @ 5:28 pm

Update of 5:20 AM ET on Sept. 29. 2022 —

The octads of the [24, 8, 8] cube-motif code
can be transformed by the permutation below
into octads recognizable, thanks to the Miracle
Octad Generator (MOG) of R. T. Curtis, as
belonging to the Golay code.

Comments Off

Thursday, September 8, 2022

Analogy in Mathematics: Chevron Variations

Filed under: General — Tags: Analogy and Mysticism, Chevron Variations, Circle Square Gauss, Maltese V, Octad, The 24 Variations — m759 @ 1:51 am

André Weil in 1940 on analogy in mathematics —

. "Once it is possible to translate any particular proof from one theory to another, then the analogy has ceased to be productive for this purpose; it would cease to be at all productive if at one point we had a meaningful and natural way of deriving both theories from a single one. In this sense, around 1820, mathematicians (Gauss, Abel, Galois, Jacobi) permitted themselves, with anguish and delight, to be guided by the analogy between the division of the circle (Gauss’s problem) and the division of elliptic functions. Today, we can easily show that both problems have a place in the theory of abelian equations; we have the theory (I am speaking of a purely algebraic theory, so it is not a matter of number theory in this case) of abelian extensions. Gone is the analogy: gone are the two theories, their conflicts and their delicious reciprocal reflections, their furtive caresses, their inexplicable quarrels; alas, all is just one theory, whose majestic beauty can no longer excite us. Nothing is more fecund than these slightly adulterous relationships; nothing gives greater pleasure to the connoisseur, whether he participates in it, or even if he is an historian contemplating it retrospectively, accompanied, nevertheless, by a touch of melancholy. The pleasure comes from the illusion and the far from clear meaning; once the illusion is dissipated, and knowledge obtained, one becomes indifferent at the same time; at least in the Gitâ there is a slew of prayers (slokas) on the subject, each one more final than the previous ones."

"The pleasure comes from the illusion" . . .

Exercise:

Compare and contrast the following structure with the three
"bricks" of the R. T. Curtis Miracle Octad Generator (MOG).

Note that the 4-row-2-column "brick" at left is quite
different from the other two bricks, which together
show chevron variations within a Galois tesseract —

Comments Off

Sunday, September 4, 2022

Dice and the Eightfold Cube

Filed under: General — Tags: Clever, Cube School, Octad — m759 @ 4:47 pm

At Hiroshima on March 9, 2018, Aitchison discussed another
"hexagonal array" with two added points… not at the center, but
rather at the ends of a cube's diagonal axis of symmetry.

See some related illustrations below.

Fans of the fictional "Transfiguration College" in the play
"Heroes of the Fourth Turning" may recall that August 6,
another Hiroshima date, was the Feast of the Transfiguration.

Iain Aitchison's 'dice-labelled' cuboctahedron at Hiroshima, March 2018

The exceptional role of 0 and ∞in Aitchison's diagram is echoed
by the occurence of these symbols in the "knight" labeling of a
Miracle Octad Generator octad —

Transposition of 0 and ∞in the knight coordinatization
induces the symplectic polarity of PG(3,2) discussed by
(for instance) Anne Duncan in 1968.

Comments Off

Friday, August 19, 2022

The Guy Embedding

Filed under: General — Tags: Octad, on220819, The Guy Embedding — m759 @ 3:43 pm

M. J. T. Guy discovered that the lexicographic version
of the Golay code contains, embedded within it, the
Miracle Octad Generator (MOG) of R. T. Curtis.

For 12 basis vectors of the lexicographic version, see below.

For some context, click the embedded guy.

For a closely related, but simpler, mathematical
structure, see posts tagged The Omega Matrix.

Comments Off

Saturday, February 5, 2022

Mathieu Cube Labeling

Filed under: General — Tags: Aitchison, Cube Labeling, Geometric Theology, Mathieu Cube, Octad — m759 @ 2:08 pm

Shown below is an illustration from "The Puzzle Layout Problem" —

September 2003
Lecture Notes in Computer Science 2912:500-501
DOI:10.1007/978-3-540-24595-7_50
Source: DBLP
Conference: Graph Drawing, 11th International Symposium,
GD 2003, Perugia, Italy, September 21-24, 2003, Revised Papers
Authors: Kozo Sugiyama, Seok-Hee Hong, Atsuhiko Maeda

Exercise: Using the above numerals 1 through 24
(with 23 as 0 and 24 as ∞) to represent the points
∞, 0, 1, 2, 3 … 22 of the projective line over GF(23),
reposition the labels 1 through 24 in the above illustration
so that they appropriately* illustrate the cube-parts discussed
by Iain Aitchison in his March 2018 Hiroshima slides on
cube-part permutations by the Mathieu group M₂₄.

A note for Northrop Frye —

Interpenetration in the eightfold cube — the three midplanes —

A deeper example of interpenetration:

Aitchison has shown that the Mathieu group M₂₄ has a natural
action on the 24 center points of the subsquares on the eightfold
cube's six faces (four such points on each of the six faces). Thus
the 759 octads of the Steiner system S(5, 8, 24) interpenetrate
on the surface of the cube.

* "Appropriately" — I.e. , so that the Aitchison cube octads correspond
exactly, via the projective-point labels, to the Curtis MOG octads.

Comments Off

Thursday, February 3, 2022

Through the Asian Looking Glass

Filed under: General — Tags: Knight Move, Octad, Octad Generator, The Guy Embedding — m759 @ 10:59 pm

The Miracle Octad Generator (MOG) —
The Conway-Sloane version of 1988:

Embedding Change, Illustrated

See also the 1976 R. T. Curtis version, of which the Conway-Sloane version
is a mirror reflection —

“There is a correspondence between the two systems
of 35 groups, which is illustrated in Fig. 4 (the MOG or
Miracle Octad Generator).”
—R.T. Curtis, “A New Combinatorial Approach to M₂₄,”
Mathematical Proceedings of the Cambridge Philosophical
Society (1976), 79: 25-42

Curtis’s 1976 Fig. 4. (The MOG.)

The Guy Embedding (named for M.J.T., not Richard K., Guy) states that
the MOG is naturally embedded in the codewords of the extended binary
Golay code, if those codewords are generated in lexicographic order.

The above reading order for the MOG 4×6 array —
down the columns, from left to right — yields the Conway-Sloane MOG.

Since that is a mirror image of the original Curtis MOG, the reading order
yielding that MOG is down the columns, from right to left.

"Traditionally, Chinese, Japanese, Vietnamese and Korean are written vertically
in columns going from top to bottom and ordered from right to left, with each
new column starting to the left of the preceding one." — Wikipedia

The Asian reading order has certain artistic advantages:

Comments Off

Four-Color Structures (Review)

Filed under: General — Tags: Four Color, Heartland Sutra, Octad, Octad Generator — m759 @ 1:30 pm

Miracle Octad Generator — Analysis of Structure

For those who prefer art that is less abstract — Heartland Sutra.

Comments Off

Saturday, January 29, 2022

On the Diamond-Theorem Group* of Order 322,560

Filed under: General — Tags: Embedding Change, Kummerhenge, Octad, Overarching Group, Priority, Strange Change — m759 @ 2:06 am

Taormina and Wendland have often discussed this group, which they
call "overarching" within the context of their Mathieu-moonshine research.

This seems to be the first time they have attempted to explore its geometric
background as an affine group, apart from its role as "the octad group" in the
researches of R. T. Curtis and John Conway on the large Mathieu group M₂₄.

* See a Log24 post of June 1, 2013.

Comments Off

Sunday, January 2, 2022

Annals of Modernism: UR–Grid

Filed under: General — Tags: Octad, Octad Generator — m759 @ 10:09 am

The above New Yorker art illustrates the 2×4 structure of
an octad in the Miracle Octad Generator of R. T. Curtis.

Enthusiasts of simplicity may note how properties of this eight-cell
2×4 grid are related to those of the smaller six-cell 3×2 grid:

See Nocciolo in this journal and . . .

Further reading on the six-set – eight-set relationship:

the diamond theorem correlation.

Comments Off

Thursday, December 30, 2021

Antidote to Chaos?

Filed under: General — Tags: 2x4, Octad, Octad Generator, Ogdoad Space, Yale Art Notes — m759 @ 3:57 pm

Some formal symmetry —

"… each 2×4 "brick" in the 1974 Miracle Octad Generator
of R. T. Curtis may be constructed by folding a 1×8 array
from Turyn's 1967 construction of the Golay code.

Folding a 2×4 Curtis array yet again yields
the 2x2x2 eightfold cube ."

— Steven H. Cullinane on April 19, 2016 — The Folding.

Related art-historical remarks:

The Shape of Time (Kubler, Yale U.P., 1962).

See yesterday's post The Thing .

Comments Off

Friday, December 17, 2021

Group Actions

Filed under: General — Tags: Octad — m759 @ 5:32 am

The above title might describe the long damned nightmare
that is the history of the human species, or — a usage I prefer —
a concept from pure mathematics. For an example of the latter,
see posts tagged Octad Group and the URL http://octad.group.

Comments Off

Thursday, December 16, 2021

Veritas

Filed under: General — Tags: Octad, QDOS — m759 @ 9:29 am

"You get right down to the naked truth
With those dirty, dirty looks" — Juice Newton (1983)

Search result for "Sandringham juice octads" —

Comments Off

Friday, October 8, 2021

Square Dance, 1979-2021

Filed under: General — Tags: Gaberdiel, Mathieu Moonshine, Octad, Octad Generator, Octad Group, Overarching, Overarching Group — m759 @ 9:45 am

Comments Off

Sunday, May 30, 2021

Framed

Filed under: General — Tags: Octad, Octad Generator — m759 @ 10:55 pm

Related reading: "Frame Analysis" in this journal.

Update of Monday, May 31, 2021 —

For connoisseurs of bullshit —

"… it would have made for a memorable
photograph, the image preserved within
he confines of a four-by-six-inch print."

— Lincoln Perry on a remembered scene
in "If You Frame It Like That," an essay in
The American Scholar dated March 2, 2020,
linked to today at Arts & Letters Daily
(A website whose motto is VERITAS ODIT
MORAS , "Truth hates delay").

And then there is non-bullshit about a
four-by-six frame —

Bullshit addicts pondering the meaning of the letter "Q" may consult
"Q is for Quelle ," "Q is for Quality," and this journal on the above
American Scholar date — March 2, 2020.

Comments Off

Saturday, May 8, 2021

A Tale of Two Omegas

Filed under: General — Tags: Octad, Octad Generator — m759 @ 5:00 am

The Greek capital letter Omega, Ω, is customarily
used to denote a set that is acted upon by a group.
If the group is the affine group of 322,560
transformations of the four-dimensional
affine space over the two-element Galois field,
the appropriate Ω is the 4×4 grid above.

See the Cullinane diamond theorem .

If the group is the large Mathieu group of
244,823,040 permutations of 24 things,
the appropriate Ω is the 4×6 grid below.

See the Miracle Octad Generator of R. T. Curtis.

Comments Off

Thursday, April 22, 2021

A New Concrete Model for an Old Abstract Space

Filed under: General — Tags: At box759 site, Octad, Square Model — m759 @ 4:31 am

The April 20 summary I wrote for ScienceOpen.com suggests
a different presentation of an Encyclopedia of Mathematics
article from 2013 —

(Click to enlarge.)

Keywords: PG(3,2), Fano space, projective space, finite geometry, square model,
Cullinane diamond theorem, octad group, MOG.

Cite as

Cullinane, Steven H. (2021).
“The Square Model of Fano’s 1892 Finite 3-Space.”
Zenodo. https://doi.org/10.5281/zenodo.4718182 .

An earlier version of the square model of PG(3,2) —

Comments Off

Monday, April 19, 2021

Diamond Theorem at ScienceOpen

Filed under: General — Tags: Gutter Filter, Octad, Square Model — m759 @ 1:22 pm

Update on April 20, 2021 —
The following was added today to the above summary:

“It describes a group of 322,560 permutations, later known as
‘the octad group,’ that now plays a role in speculative high-energy physics.
See Moonshine, Superconformal Symmetry, and Quantum Error Correction .”

Comments Off

Sunday, March 21, 2021

Mathematics and Narrative: The Unity

Filed under: General — Tags: Cube School, Octad, Octad Generator, The Fano Hallows — m759 @ 10:25 pm

“To conquer, three boxes* have to synchronize and join together into the Unity.”

―Wonder Woman in Zack Snyder’s Justice League

* Cf. Aitchison’s Octads —

Comments Off

Thursday, February 25, 2021

Compare and Contrast

Filed under: General — Tags: Cube School, Octad, Octad Generator, The Guy Embedding — m759 @ 12:31 pm

“… What is your dream—your ideal? What is your News from Nowhere,
or, rather, What is the result of the little shake your hand has given to
the old pasteboard toy with a dozen bits of colored glass for contents?
And, most important of all, can you present it in a narrative or romance
which will enable me to pass an idle hour not disagreeably? How, for instance,
does it compare in this respect with other prophetic books on the shelf?”

— Hudson, W. H.. A Crystal Age (p. 2). Open Road Media. Kindle Edition.

See as well . . .

The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.

Comments Off

Wednesday, January 27, 2021

Adoration of the Cube . . .

Filed under: General — Tags: Cube Monstrance, Myth and Memory, Octad, Octad Generator — m759 @ 2:53 am

Continues.

Related vocabulary —

See as well the word facet in this journal.

Analogously, one might write . . .

A Hiroshima cube consists of 6 faces ,
each with 4 squares called facets ,
for a total of 24 facets. . . ."

(See Aitchison's Octads , a post of Feb. 19, 2020.)

Click image to enlarge. Background: Posts tagged 'Aitchison.'"

Comments Off

Thursday, December 3, 2020

Brick Joke

Filed under: General — Tags: Brick Symmetry, Kummerhenge, Octad, Octad Generator, Octad Group, Overarching Group, The Lindelof Watchmen — m759 @ 10:56 am

The "bricks" in posts tagged Octad Group suggest some remarks
from last year's HBO "Watchmen" series —

Related material — The two bricks constituting a 4×4 array, and . . .

"(this is the famous Kummer abstract configuration )"
— Igor Dolgachev, ArXiv, 16 October 2019.

As is this —

The phrase "octad group" does not, as one might reasonably
suppose, refer to symmetries of an octad (a "brick"), but
instead to symmetries of the above 4×4 array.

A related Broomsday event for the Church of Synchronology —

Comments Off

Wednesday, October 7, 2020

Between Pyramid and Pentagon

Filed under: General — Tags: Doily, Octad, Octad Generator — m759 @ 12:34 am

The 35 small squares between the pyramid and the pentagon
in the above search result illustrate the role of finite geometry
in the Miracle Octad Generator of R. T. Curtis.

'Then a miracle occurs' cartoon

Cartoon by S. Harris

Comments Off

Monday, September 28, 2020

Acceptance

Filed under: General — Tags: Octad — m759 @ 10:30 am

Also on May 2, 2020 — A paper on Cubism as Religion is accepted:

Some may question the desirability of acceptance by MDPI.
Acceptance at the Pearly Gates is another matter.

Comments Off

Wednesday, September 2, 2020

Space Wars: Sith Pyramid vs. Jedi Cube

Filed under: General — Tags: Octad, Octad Generator — m759 @ 11:18 am

For the Sith Pyramid, see posts tagged Pyramid Game.
For the Jedi Cube, see posts tagged Enigma Cube
and cube-related remarks by Aitchison at Hiroshima.

This post was suggested by two events of May 16, 2019 —
A weblog post by Frans Marcelis on the Miracle Octad
Generator of R. T. Curtis (illustrated with a pyramid),
and the death of I. M. Pei, architect of the Louvre pyramid.

That these events occurred on the same date is, of course,
completely coincidental.

Perhaps Dan Brown can write a tune to commemorate
the coincidence.

Comments Off

Thursday, August 27, 2020

The Complete Extended Binary Golay Code

Filed under: General — Tags: Canonicity, MJT Guy, Octad, Octad Generator, Springer, The Guy Embedding — m759 @ 12:21 pm

All 4096 vectors in the code are at . . .

http://neilsloane.com/oadir/oa.4096.12.2.7.txt.

Sloane’s list* contains the 12 generating vectors
listed in 2011 by Adlam —

As noted by Conway in Sphere Packings, Lattices and Groups ,
these 4096 vectors, constructed lexicographically, are exactly
the same vectors produced by using the Conway-Sloane version
of the Curtis Miracle Octad Generator (MOG). Conway says this
lexico-MOG equivalence was first discovered by M. J. T. Guy.

(Of course, any permutation of the 24 columns above produces
a version of the code qua code. But because the lexicographic and
the MOG constructions yield the same result, that result is in
some sense canonical.)

See my post of July 13, 2020 —

The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.

For some related results, Google the twelfth generator:

* Sloane’s list is of the codewords as the rows of an orthogonal array —

Wednesday, August 5, 2020

Multifaceted Unities

Filed under: General — Tags: Dramarama, Facets, Natural Diagram, Octad — m759 @ 10:45 am

Facettenreiche Grundlage:

Multifaceted Foundation: Facettenreiche Grundlage

Comments Off

Tuesday, July 28, 2020

For 7/28

Filed under: General — Tags: Octad, Octad Generator — m759 @ 8:33 am

Miracle Octad Generator — Analysis of Structure

Comments Off

Friday, July 17, 2020

Theoretical as Well as Practical Value

Filed under: General — Tags: Devs, Natural Diagram, Octad, Octad Generator, Structure and Mutability — m759 @ 12:25 am

See also The Lexicographic Octad Generator (LOG) (July 13, 2020)
and Octads and Geometry (April 23, 2020).

Comments Off

Friday, May 22, 2020

Annals of Crystalline Beauty

Filed under: General — Tags: Octad — m759 @ 4:58 pm

The phrase “laborious cerebration” quoted in the previous post,
Sombre Figuration, suggests . . .

For an example of such cerebration, see Aitchison’s Octads.

Comments Off

Sunday, May 17, 2020

“The Ultimate Epistemological Fact”

Filed under: General — Tags: Eightfold Cube, Geometric Theology, Octad, Ogdoad Space — m759 @ 11:49 pm

"Let me say this about that." — Richard Nixon

Interpenetration in Weyl's epistemology —

Interpenetration in Mazzola's music theory —

Interpenetration in the eightfold cube — the three midplanes —

A deeper example of interpenetration:

Comments Off

Thursday, May 7, 2020

Kant as Diamond Cutter

Filed under: General — Tags: MSRI Spring, Octad, The Fano Hallows — m759 @ 4:26 am

“He wished Kant were alive. Kant would have appreciated it.
That master diamond cutter.”

— Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance , Part III.

Kant’s “category theory” —

“In the Transcendental Analytic, Kant deduces the table of twelve categories, or pure concepts of the understanding….

The categories must be ‘schematized’ because their non-empirical origin in pure understanding prevents their having the sort of sensible content that would connect them immediately to the objects of experience; transcendental schemata are mediating representations that are meant to establish the connection between pure concepts and appearances in a rule-governed way. Mathematical concepts are discussed in this context since they are unique in being pure but also sensible concepts: they are pure because they are strictly a priori in origin, and yet they are sensible since they are constructed in concreto . ”

— Shabel, Lisa, “Kant’s Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Spring 2016 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/spr2016/entries/kant-mathematics/>.

See also The Diamond Theorem and Octad.us.

Comments Off

Monday, May 4, 2020

The Man Behind the Counter

Filed under: General — Tags: Octad, The Next Level — m759 @ 7:06 pm

The title is a phrase from the Suzanne Vega song in the previous post.

“Always busy counting . . . .” — Tagline at Peter J. Cameron’s weblog.

“This morning brought the news that Jan Saxl died on Saturday.”

— Peter J. Cameron today

A search for Saxl in this weblog yields a post related to a topic in
Wolfram Neutsch’s book Coordinates. See Saturday’s post
“Turyn’s Octad Theorem: The Next Level.”

Related narrative from the Saturday post —

Related narrative from Sunday’s Westworld finale —

Comments Off

Wednesday, April 29, 2020

Curtis at Pilsen, Thursday, July 5, 2018

Filed under: General — Tags: Hanuman, Kummerhenge, Mythspace, Octad, Octad Generator, Priority — m759 @ 11:48 am

For an account by R. T. Curtis of how he discovered the Miracle Octad Generator,
see slides by Curtis, “Graphs and Groups,” from his talk on July 5, 2018, at the
Pilsen conference on algebraic graph theory, “Symmetry vs. Regularity: The first
50 years since Weisfeiler-Leman stabilization” (WL2018).

See also “Notes to Robert Curtis’s presentation at WL2018,” by R. T. Curtis.

Meanwhile, here on July 5, 2018 —

Simultaneous perspective does not look upon language as a path because it is not the search for meaning that orients it. Poetry does not attempt to discover what there is at the end of the road; it conceives of the text as a series of transparent strata within which the various parts—the different verbal and semantic currents—produce momentary configurations as they intertwine or break apart, as they reflect each other or efface each other. Poetry contemplates itself, fuses with itself, and obliterates itself in the crystallizations of language. Apparitions, metamorphoses, volatilizations, precipitations of presences. These configurations are crystallized time: although they are perpetually in motion, they always point to the same hour—the hour of change. Each one of them contains all the others, each one is inside the others: change is only the oft-repeated and ever-different metaphor of identity.

— Paz, Octavio. The Monkey Grammarian
(Kindle Locations 1185-1191).
Arcade Publishing. Kindle Edition.

The 2018 Log24 post containing the above Paz quote goes on to quote
remarks by Lévi-Strauss. Paz’s phrase “series of transparent strata”
suggests a review of other remarks by Lévi-Strauss in the 2016 post
“Key to All Mythologies.“

Comments Off

Monday, April 27, 2020

The Cracked Nut

Filed under: General — Tags: Art Issue, Octad, The Nutshell Suite — m759 @ 1:25 pm

“At that instant he saw, in one blaze of light, an image of unutterable
conviction, the reason why the artist works and lives and has his being –
the reward he seeks –the only reward he really cares about, without which
there is nothing. It is to snare the spirits of mankind in nets of magic,
to make his life prevail through his creation, to wreak the vision of his life,
the rude and painful substance of his own experience, into the congruence
of blazing and enchanted images that are themselves the core of life, the
essential pattern whence all other things proceed, the kernel of eternity.”

— Thomas Wolfe, Of Time and the River

“… the stabiliser of an octad preserves the affine space structure on its
complement, and (from the construction) induces AGL(4,2) on it.
(It induces A₈ on the octad, the kernel of this action being the translation
group of the affine space.)”

— Peter J. Cameron,
The Geometry of the Mathieu Groups (pdf)

“The yarns of seamen have a direct simplicity, the whole meaning
of which lies within the shell of a cracked nut. But Marlow was not
typical (if his propensity to spin yarns be excepted), and to him the
meaning of an episode was not inside like a kernel but outside…."

— Joseph Conrad in Heart of Darkness

Comments Off

Sunday, March 22, 2020

Eightfold Site

Filed under: General — Tags: Eightfold Cube, Octad — m759 @ 2:00 am

A brief summary of the eightfold cube is now at octad.us.

Comments Off

Monday, December 23, 2019

Orbit

Filed under: General — Tags: Inner-Space Variations, Obit et Orbit, Octad — m759 @ 7:34 pm

"December 22, the birth anniversary of India’s famed mathematician
Srinivasa Ramanujan, is celebrated as National Mathematics Day."
— Indian Express yesterday

"Orbits and stabilizers are closely related." — Wikipedia

Symmetries by Plato and R. T. Curtis —

In the above, 322,560 is the order
of the octad stabilizer group .

Comments Off

Saturday, December 14, 2019

Colorful Tale

Filed under: General — Tags: Eightfold Cube, Four Color, Interality, Octad, Octad Generator — m759 @ 9:00 pm

(Continued)

The above image is from

"A Four-Color Theorem:
Function Decomposition Over a Finite Field,"
http://finitegeometry.org/sc/gen/mapsys.html.

These partitions of an 8-set into four 2-sets
occur also in Wednesday night's post
Miracle Octad Generator Structure.

This post was suggested by a Daily News
story from August 8, 2011, and by a Log24
post from that same date, "Organizing the
Mine Workers" —

Comments Off

Thursday, October 31, 2019

56 Triangles

Filed under: General — Tags: Geometry of Even Subsets, Octad, Octad Generator, Triangles-Spreads-Mathieu — m759 @ 8:09 am

The post “Triangles, Spreads, Mathieu” of October 29 has been
updated with an illustration from the Curtis Miracle Octad Generator.

Related material — A search in this journal for “56 Triangles.”

Comments Off

Friday, October 25, 2019

Facettenreiche Gestaltung

Filed under: General — Tags: Facets, Octad, Springer — m759 @ 12:31 pm

On the word Gestaltung —

(Here “eidolon” should instead be “eidos .”)

A search for a translation of the book "Facettenreiche Mathematik " —

A paper found in the above search —

A related translation —

Thursday, October 3, 2019

Apocalypse* Note

Filed under: General — Tags: Aitchison, Octad — m759 @ 7:00 pm

For a first look at octad.space, see that domain.
For a second look, see octad.design.
For some other versions, see Aitchison in this journal.

* The X-Men character.

Comments Off

Saturday, September 21, 2019

Annals of Random Fandom

Filed under: General — Tags: Octad, Octad Generator, Pyramid Game, Tetrahedron vs. Square — m759 @ 5:46 pm

For Dan Brown fans …

… and, for fans of The Matrix, another tale
from the above death date: May 16, 2019 —

An illustration from the above
Miracle Octad Generator post:

Related mathematics — Tetrahedron vs. Square.

Comments Off

Thursday, July 25, 2019

Simple

Filed under: General — Tags: Octad, Octad Generator — m759 @ 2:08 pm

From "110 in the Shade" —

A quote from "Marshall, Meet Bagger," July 29, 2011:

"Time for you to see the field."

_________________________________________________________

From a Log24 search for "To See the Field" —

For further details, see the 1985 note
"Generating the Octad Generator."

Comments Off

Wednesday, July 24, 2019

The Batty Farewell

Filed under: General — Tags: Octad, Octad Generator, Tears in Rain — m759 @ 8:40 pm

Adam Rogers today on "Rutger Hauer in Blade Runner , playing
the artificial* person Roy Batty in his death scene."

* See the word "Artifice" in this journal,
as well as Tears in Rain . . .

Game Over

Coordinates for generating the Miracle Octad Generator

… and Adam Rogers in
the previous post.

Comments Off

Frameworks

Filed under: General — Tags: Octad, Octad Generator — m759 @ 7:50 pm

"Games provide frameworks that miniaturize
and represent idealized realities; so do narratives."

— Adam Rogers, Sunday, July 21, 2019, at Wired

Reviewing yesterday's post Word Magic —

See also a technological framework (the microwave at left) vs. a
purely mathematical framework (the pattern on the towel at right)
in the image below:

For some backstory about the purely mathematical framework,
see Octad Generator in this journal.

Comments Off

Saturday, May 4, 2019

The Chinese Jars of Shing-Tung Yau

Filed under: General — Tags: Aitchison, Master Plan, Octad, Octad Generator, The Crimson Possibilities — m759 @ 11:00 am

The title refers to Calabi-Yau spaces.

T. S. Eliot —

Four Quartets

. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.

A less "cosmic" but still noteworthy code — The Golay code.

This resides in a 12-dimensional space over GF(2).

Related material from Plato and R. T. Curtis —

A related Calabi-Yau "Chinese jar" first described in detail in 1905 —

A figure that may or may not be related to the 4x4x4 cube that
holds the classical Chinese "cosmic code" — the I Ching —

ftp://ftp.cs.indiana.edu/pub/hanson/forSha/AK3/old/K3-pix.pdf

Comments Off

Wednesday, May 1, 2019

The Medium and the Message

Filed under: General — Tags: McLuhan and Mathematics, Octad, Octad Generator — m759 @ 6:45 pm

In memory of Quentin Fiore — from a Log24 search for McLuhan,
an item related to today's previous post . . .

Related material from Log24 on the above-reported date of death —

See also, from a search for Analogy in this journal . . .

Comments Off

Friday, April 5, 2019

April 1 Omega

Filed under: General — Tags: Octad, Springer — m759 @ 10:58 pm

From posts tagged Number Art —

From the novel Point Omega —

Related material for
Mathematics Awareness Month —

Also on 07/18/2015 —

Comments Off

Thursday, March 28, 2019

Eight and Seven

Filed under: General — Tags: Number Art, Octad — m759 @ 8:56 am

Comments Off

Saturday, March 16, 2019

Multifaceted Narrative

Filed under: General — Tags: Facets, Jencks, Octad — m759 @ 2:40 pm

"Here, modernism is defined as an autonomous body
of ideas, having little or no outward reference, placing
considerable emphasis on formal aspects of the work
and maintaining a complicated—indeed, anxious—
rather than a naïve relationship with the day-to-day
world, which is the de facto view of a coherent group
of people, such as a professional or discipline-based
group that has a high sense of the seriousness and
value of what it is trying to achieve. This brisk definition…."

— Jeremy Gray, Plato's Ghost: The Modernist
Transformation of Mathematics , Princeton, 2008

"Even as the dominant modernist narrative was being written,
there were art historians who recognized that it was inaccurate.
The narrative was too focused on France . . . . Nor was it
correct to build the narrative so exclusively around formalism;
modernism was far messier, far more multifaceted than that."

— Jane Kallir, https://www.tabletmag.com/
jewish-arts-and-culture/visual-art-and-design/
269564/the-end-of-middle-class-art,
quoted here on the above date — Sept. 11, 2018.

From some related Log24 posts —

Comments Off

Thursday, March 7, 2019

In Reality

Filed under: General — Tags: Octad, Octad Generator — m759 @ 11:45 am

The previous post, quoting a characterization of the R. T. Curtis
Miracle Octad Generator , describes it as a "hand calculator ."

Other views

A "natural diagram " —

A geometric object —

Comments Off

Tuesday, March 5, 2019

A Block Design 3-(16,4,1) as a Steiner Quadruple System:

Filed under: General — Tags: Helsinki Math, Octad, Octad Generator, Wikipedia Scholarship — m759 @ 11:19 am

A Midrash for Wikipedia

Midrash —

Related material —

________________________________________________________________________________

Comments Off

Friday, March 1, 2019

Wikipedia Scholarship (Continued)

Filed under: General — Tags: Helsinki Math, Octad, Octad Generator, Priority, Wikipedia Scholarship — m759 @ 12:45 pm

This post continues a post from yesterday on the square model of
PG(3,2) that apparently first appeared (presented as such*) in . . .

Cullinane, "Symmetry invariance in a diamond ring,"
Notices of the AMS , pp. A193-194, Feb. 1979.

Yesterday's Wikipedia presentation of the square model was today
revised by yet another anonymous author —

Revision history accounting for the above change from yesterday —

The jargon "rm OR" means "remove original research."

The added verbiage about block designs is a smokescreen having
nothing to do with the subject, which is square representation
of the 35 points and lines.

* The 35 squares, each consisting of four 4-element subsets, appeared earlier
in the Miracle Octad Generator (MOG) of R. T. Curtis (published in 1976).
They were not at that time presented as constituting a finite geometry,
either affine (AG(4,2)) or projective (PG(3,2)).

Comments Off

Friday, February 22, 2019

Back Issues of AMS Notices

Filed under: General — Tags: Octad, Octad Generator — m759 @ 3:04 pm

From the online home page of the new March issue —

Feb. 22, 2019 — AMS Notices back issues now available.

For instance . . .

Related material now at Wikipedia —

Comments Off

Thursday, February 21, 2019

Frenkel on “the Rashomon Effect”

Filed under: General — Tags: Facets, Frenkel-Metaphors, Octad, The Donen Code — m759 @ 1:44 pm

Earlier in Frenkel's above opinion piece —

"What this research implies is that we are not just hearing
different 'stories' about the electron, one of which may be
true. Rather, there is one true story, but it has many facets,
seemingly in contradiction, just like in 'Rashomon.'
There is really no escape from the mysterious — some
might say, mystical — nature of the quantum world."

See also a recent New Yorker version of the fashionable cocktail-party
phrase "the Rashomon effect."

For a different approach to the dictum "there is one true story, but
it has many facets," see . . .

"Read something that means something."
— New Yorker motto

Comments Off

Thursday, February 7, 2019

Geometry of the 4×4 Square: The Kummer Configuration

Filed under: General — Tags: Kummerhenge, Octad, Octad Generator, The Omega Matrix — m759 @ 12:00 am

From the series of posts tagged Kummerhenge —

A Wikipedia article relating the above 4×4 square to the work of Kummer —

A somewhat more interesting aspect of the geometry of the 4×4 square
is its relationship to the 4×6 grid underlying the Miracle Octad Generator
(MOG) of R. T. Curtis. Hudson's 1905 classic Kummer's Quartic Surface
deals with the Kummer properties above and also foreshadows, without
explicitly describing, the finite-geometry properties of the 4×4 square as
a finite affine 4-space — properties that are of use in studying the Mathieu
group M₂₄with the aid of the MOG.

Comments Off

Friday, January 18, 2019

The Woke Grids …

Filed under: General — Tags: Dreaming Jewels, Namespace, Octad, Toy Story — m759 @ 10:45 am

… as opposed to The Dreaming Jewels .

A July 2014 Amsterdam master's thesis on the Golay code
and Mathieu group —

"The properties of G₂₄ and M₂₄ are visualized by
four geometric objects: the icosahedron, dodecahedron,
dodecadodecahedron, and the cubicuboctahedron."

Some "geometric objects" — rectangular, square, and cubic arrays —
are even more fundamental than the above polyhedra.

A related image from a post of Dec. 1, 2018 —

Comments Off

Sunday, December 2, 2018

Symmetry at Hiroshima

Filed under: G-Notes,General,Geometry — Tags: Aitchison, Geometric Theology, Mathieu Cube, Octad, Octad Generator — m759 @ 6:43 am

A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018 —

http://www.math.sci.hiroshima-u.ac.jp/
branched/files/2018/abstract/Aitchison.txt

Iain AITCHISON

Title:

Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II

Abstract:

Since antiquity, some mathematical objects have played a special role, underpinning new mathematics as understanding deepened. Perhaps archetypal are the Platonic polyhedra, subsequently related to Platonic idealism, and the contentious notion of existence of mathematical reality independent of human consciousness.

Exceptional or unique objects are often associated with symmetry – manifest or hidden. In topology and geometry, we have natural base points for the moduli spaces of closed genus 2 and 3 surfaces (arising from the 2-fold branched cover of the sphere over the 6 vertices of the octahedron, and Klein's quartic curve, respectively), and Bring's genus 4 curve arises in Klein's description of the solution of polynomial equations of degree greater than 4, as well as in the construction of the Horrocks-Mumford bundle. Poincare's homology 3-sphere, and Kummer's surface in real dimension 4 also play special roles.

In other areas: we have the exceptional Lie algebras such as E8; the sporadic finite simple groups; the division algebras: Golay's binary and ternary codes; the Steiner triple systems S(5,6,12) and S(5,8,24); the Leech lattice; the outer automorphisms of the symmetric group S6; the triality map in dimension 8; and so on. We also note such as: the 27 lines on a cubic, the 28 bitangents of a quartic curve, the 120 tritangents of a sextic curve, and so on, related to Galois' exceptional finite groups PSL2(p) (for p= 5,7,11), and various other so-called `Arnol'd Trinities'.

Motivated originally by the `Eightfold Way' sculpture at MSRI in Berkeley, we discuss inter-relationships between a selection of these objects, illustrating connections arising via highly symmetric Riemann surface patterns. These are constructed starting with a labeled polygon and an involution on its label set.

Necessarily, in two lectures, we will neither delve deeply into, nor describe in full, contexts within which exceptional objects arise. We will, however, give sufficient definition and detail to illustrate essential inter-connectedness of those exceptional objects considered.

Our starting point will be simplistic, arising from ancient Greek ideas underlying atomism, and Plato's concepts of space. There will be some overlap with a previous talk on this material, but we will illustrate with some different examples, and from a different philosophical perspective.

Some new results arising from this work will also be given, such as an alternative graphic-illustrated MOG (Miracle Octad Generator) for the Steiner system S(5,8,24), and an alternative to Singerman – Jones' genus 70 Riemann surface previously proposed as a completion of an Arnol'd Trinity. Our alternative candidate also completes a Trinity whose two other elements are Thurston's highly symmetric 6- and 8-component links, the latter related by Thurston to Klein's quartic curve.

See also yesterday morning's post, "Character."

Update: For a followup, see the next Log24 post.

Comments Off

Saturday, December 1, 2018

Character

Filed under: General — Tags: Aitchison, Octad — m759 @ 11:00 am

"What we do may be small, but it has
a certain character of permanence."

— G. H. Hardy,
A Mathematician's Apology

Comments Off

Wednesday, October 3, 2018

Adamantine Meditation

Filed under: General,Geometry — Tags: Octad — m759 @ 12:24 pm

A Catholic philosopher —

Related art —

Comments Off

Sunday, September 23, 2018

Three Times Eight

Filed under: General,Geometry — Tags: ABC Art 2018, Octad, Octad Generator — m759 @ 9:21 am

The New York Times 's Sunday School today —

I prefer the three bricks of the Miracle Octad Generator —

Comments Off

Wednesday, August 8, 2018

8/8

Filed under: General,Geometry — Tags: Alperin, D8ing, D8ing Horseness, Diamond Theorem Correlation, Horseness, Octad — m759 @ 6:00 am

From mathoverflow.net on Dec. 7, 2016 —

The exceptional isomorphism between
PGL(3,2) and PSL(2,7): geometric origin?

Essentially the same question was asked earlier at

math.stackexchange.com on Aug. 2, 2010.

See also this journal in November 2017 —

"Read something that means something."
— New Yorker ad

Background — Relativity Problem in Log24.

Comments Off

Wednesday, July 18, 2018

Doodle

Filed under: General,Geometry — Tags: Octad, Octad Generator — m759 @ 12:00 pm

From "The Educated Imagination: A Website Dedicated
to Northrop Frye" —

"In one of the notebooks for his first Bible book Frye writes,

'For at least 25 years I’ve been preoccupied by
the notion of a key to all mythologies.' . . . .

Frye made a valiant effort to provide a key to all mythology,
trying to fit everything into what he called the Great Doodle. . . ."

From a different page at the same website —

Here Frye provides a diagram of four sextets.

I prefer the Miracle Octad Generator of R. T. Curtis —

Counting symmetries with the orbit-stabilizer theorem .

Comments Off

Monday, June 25, 2018

The Trials of Device

Filed under: General — Tags: Five by Five, Octad, Octad Generator — m759 @ 9:34 am

"A blank underlies the trials of device."

— Wallace Stevens

"Designing with just a blank piece of paper is very quiet."

— Kate Cullinane

Related material —

An image posted at 12 AM ET December 25, 2014:

The image stands for the
phrase "five by five,"
meaning "loud and clear."

Other posts featuring the above 5×5 square with some added structure:

Comments Off

Tuesday, November 7, 2017

Polarities and Correlation

Filed under: General,Geometry — Tags: Conceptualist Minimalism, Gopnik, Octad — m759 @ 11:00 pm

"Read something that means something."
— New Yorker ad

Comments Off

Thursday, August 24, 2017

Maori Chess, Vol. 2

Filed under: General,Geometry — Tags: Octad, Octad Generator — m759 @ 4:20 pm

This just in …

From IMDb —

From Radio New Zealand —

"Genesis Potini died of a heart attack aged 46
on the 15th August 2011."

The 15th of August in New Zealand overlapped
the 14th of August in the U.S.A.

From a Log24 post, "Sunday Review," on August 14, 2011 —

Part II (from "Marshall, Meet Bagger," July 29):

"Time for you to see the field."

For further details, see the 1985 note
"Generating the Octad Generator."

McLuhan was a Toronto Catholic philosopher.
For related views of a Montreal Catholic philosopher,
see the Saturday evening post.

Comments Off

Tuesday, May 2, 2017

Image Albums

Filed under: General,Geometry — Tags: Art Space, Eightfold Cube, Four Color, Galois's Space, Octad, Octad Generator — m759 @ 1:05 pm

Pinterest boards uploaded to the new m759.net/piwigo —

Diamond Theorem

Diamond Theorem Correlation

Miracle Octad Generator

The Eightfold Cube

Six-Set Geometry

Diamond Theory Cover

Update of May 2 —

Four-Color Decomposition

Binary Galois Spaces

The Galois Tesseract

Update of May 3 —

Desargues via Galois

The Tetrahedral Model

Solomon's Cube

Update of May 8 —

Art Space board created at Pinterest

Comments Off

Friday, February 17, 2017

Kostant Is Dead

Filed under: General,Geometry — Tags: Facets, Octad — m759 @ 1:10 pm

"Bertram Kostant, professor emeritus of mathematics at MIT,
died at the Hebrew Senior Rehabilitation Center in Roslindale,
Massachusetts, on Thursday, Feb. 2, at the age of 88."

— MIT News, story dated Feb. 16, 2017

See also a search for Kostant in this journal.

Regarding the discussions of symmetries and "facets" found in
that search —

Kostant:

“A word about E(8). In my opinion, and shared by others,
E(8) is the most magnificent ‘object’ in all of mathematics.
It is like a diamond with thousands of facets. Each facet
offering a different view of its unbelievable intricate internal
structure.”

Cullinane:

In the Steiner system S(5, 8, 24) each octad might be
regarded as a "facet," with the order of the system's
automorphism group, the Mathieu group M₂₄ , obtained
by multiplying the number of such facets, 759, by the
order of the octad stabilizer group, 322,560.

Analogously …

Platonic solids' symmetry groups

Comments Off

Wednesday, December 7, 2016

Spreads and Conwell’s Heptads

Filed under: General,Geometry — Tags: Geometry of Even Subsets, Octad, Octad Generator, Six-Set — m759 @ 7:11 pm

For a concise historical summary of the interplay between
the geometry of an 8-set and that of a 16-set that is
involved in the the Miracle Octad Generator approach
to the large Mathieu group M₂₄, see Section 2 of …

Alan R. Prince
A near projective plane of order 6 (pp. 97-105)
Innovations in Incidence Geometry
Volume 13 (Spring/Fall 2013).

This interplay, notably discussed by Conwell and
by Edge, involves spreads and Conwell’s heptads .

Update, morning of the following day (7:07 ET) — related material:

See also “56 spreads” in this journal.

Comments Off

Friday, December 2, 2016

A Small Witt Design*

Filed under: General — Tags: Lumber Room, Octad, Octad Generator — m759 @ 2:00 pm

The New York Times 's online T Magazine yesterday —

"A version of this article appears in print on December 4, 2016, on page
M263 of T Magazine with the headline: The Year of Magical Thinking."

* Thanks to Emily Witt for inadvertently publicizing the
Miracle Octad Generator of R. T. Curtis, which
summarizes the 759 octads found in the large Witt design.

Comments Off

Tuesday, September 13, 2016

Parametrizing the 4×4 Array

Filed under: General,Geometry — Tags: Geometry, Kummerhenge, Octad, Octad Generator, Priority, Six-Set — m759 @ 10:00 pm

The previous post discussed the parametrization of
the 4×4 array as a vector 4-space over the 2-element
Galois field GF(2).

The 4×4 array may also be parametrized by the symbol
0 along with the fifteen 2-subsets of a 6-set, as in Hudson's
1905 classic Kummer's Quartic Surface —

Hudson in 1905:

These two ways of parametrizing the 4×4 array — as a finite space
and as an array of 2-element sets — were related to one another
by Cullinane in 1986 in describing, in connection with the Curtis
"Miracle Octad Generator," what turned out to be 15 of Hudson's
1905 "Göpel tetrads":

A recap by Cullinane in 2013:

Click images for further details.

Comments Off

Tuesday, May 24, 2016

Rosenhain and Göpel Revisited

Filed under: General,Geometry — Tags: Dirac and Geometry, Facets, Kummerhenge, Octad, Octad Generator, Quantum Tesseract Theorem, Rosenhain and Göpel, The Omega Matrix, The Rosenhain Symmetry — m759 @ 8:23 am

The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface .

"This famous book is a prototype for the possibility
of explaining and exploring a many-faceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least,
as an everlasting symbol of mathematical culture."

— Werner Kleinert, Mathematical Reviews ,
as quoted at Amazon.com

Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4-space over
the two-element Galois field GF(2).

Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .

Some related work of my own (click images for related posts)—

Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)

Göpel tetrads as 15 of the 35 projective lines in PG(3,2)

Related terminology describing the Göpel tetrads above

Ron Shaw on symplectic geometry and a linear complex in PG(3,2)

Comments Off

Tuesday, April 19, 2016

The Folding

Filed under: General,Geometry — Tags: 2x4, Eightfold Cube, Octad, Octad Generator — m759 @ 2:00 pm

(Continued)

A recent post about the eightfold cube suggests a review of two
April 8, 2015, posts on what Northrop Frye called the ogdoad :

As noted on April 8, each 2×4 "brick" in the 1974 Miracle Octad Generator
of R. T. Curtis may be constructed by folding a 1×8 array from Turyn's
1967 construction of the Golay code.

Folding a 2×4 Curtis array yet again yields the 2x2x2 eightfold cube .

Those who prefer an entertainment approach to concepts of space
may enjoy a video (embedded yesterday in a story on theverge.com) —
"Ghost in the Shell: Identity in Space."

Comments Off

Monday, April 11, 2016

Combinatorial Spider

Filed under: General,Geometry — Tags: April 8-11 2016, Octad, Octad Generator, on160411 — m759 @ 1:16 pm

“Chaos is order yet undeciphered.”

— The novel The Double , by José Saramago,
on which the film "Enemy" was based

Some background for the 2012 Douglas Glover
"Attack of the Copula Spiders" book
mentioned in Sunday's Synchronicity Check —

"A vision of Toronto as Hell" — Douglas Glover in the
March 25, 2011, post Combinatorial Delight
For Louise Bourgeois — a post from the date of Galois's death—

For Toronto — Scene from a film that premiered there
on Sept. 8, 2013:

Comments Off

Friday, April 8, 2016

Ogdoads by Curtis

Filed under: General,Geometry — Tags: 2x4, April 8-11 2016, Octad, Octad Generator, Ogdoad Space, Seventh Seal — m759 @ 12:25 pm

As was previously noted here, the construction of the Miracle Octad Generator
of R. T. Curtis in 1974 may have involved his "folding" the 1×8 octads constructed
in 1967 by Turyn into 2×4 form.

This results in a way of picturing a well-known correspondence (Conwell, 1910)
between partitions of an 8-set and lines of the projective 3-space PG(3,2).

For some background related to the "ogdoads" of the previous post, see
A Seventh Seal (Sept. 15, 2014).

Comments Off

Monday, February 1, 2016

Religious Note

Filed under: General — Tags: Octad, Octad Generator — m759 @ 1:00 pm

Friday, November 13, 2015

A Connection between the 16 Dirac Matrices and the Large Mathieu Group

Filed under: General,Geometry — Tags: Diamond Theorem Correlation, Dirac and Geometry, Octad, Octad Generator, Quantum Tesseract Theorem, Six-Set — m759 @ 2:45 am

Note that the six anticommuting sets of Dirac matrices listed by Arfken
correspond exactly to the six spreads in the above complex of 15 projective
lines of PG(3,2) fixed under a symplectic polarity (the diamond theorem
correlation ). As I noted in 1986, this correlation underlies the Miracle
Octad Generator of R. T. Curtis, hence also the large Mathieu group.

References:

Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213-214

Cullinane, Steven H., Notes on Groups and Geometry, 1978-1986

Related material:

The 6-set in my 1986 note above also appears in a 1996 paper on
the sixteen Dirac matrices by David M. Goodmanson —

Background reading:

Ron Shaw on finite geometry, Clifford algebras, and Dirac groups
(undated compilation of publications from roughly 1994-1995)—

Comments Off

Saturday, September 19, 2015

Geometry of the 24-Point Circle

Filed under: General,Geometry — Tags: Mathieu Raumproblem, Octad, Octad Generator — m759 @ 1:13 am

The latest Visual Insight post at the American Mathematical
Society website discusses group actions on the McGee graph,
pictured as 24 points arranged in a circle that are connected
by 36 symmetrically arranged edges.

Wikipedia remarks that …

"The automorphism group of the McGee graph
is of order 32 and doesn't act transitively upon
its vertices: there are two vertex orbits of lengths
8 and 16."

The partition into 8 and 16 points suggests, for those familiar
with the Miracle Octad Generator and the Mathieu group M₂₄,
the following exercise:

Arrange the 24 points of the projective line
over GF(23) in a circle in the natural cyclic order
( ∞, 1, 2, 3, … , 22, 0 ). Can the McGee graph be
modeled by constructing edges in any natural way?

Image that may or may not be related to the extended binary Golay code and the large Witt design

In other words, if the above set of edges has no
"natural" connection with the 24 points of the
projective line over GF(23), does some other
set of edges in an isomorphic McGee graph
have such a connection?

Update of 9:20 PM ET Sept. 20, 2015:

Backstory: A related question by John Baez
at Math Overflow on August 20.

Comments Off

Thursday, August 27, 2015

Tears in the Rain

Filed under: General — Tags: Octad, Octad Generator, Tears in Rain — m759 @ 12:21 pm

For a Norwegian historian —

Game Over

Coordinates for generating the Miracle Octad Generator

Comments Off

Tuesday, March 24, 2015

Brouwer on the Galois Tesseract

Filed under: General,Geometry — Tags: Galois's Space, Octad, Priority — m759 @ 12:00 pm

Yesterday's post suggests a review of the following —

Andries Brouwer, preprint, 1982:

"The Witt designs, Golay codes and Mathieu groups"
(unpublished as of 2013)

Pages 8-9:

Substructures of S(5, 8, 24)

An octad is a block of S(5, 8, 24).

Theorem 5.1

Let B₀ be a fixed octad. The 30 octads disjoint from B₀
form a self-complementary 3-(16,8,3) design, namely

the design of the points and affine hyperplanes in AG(4, 2),
the 4-dimensional affine space over F₂.

Proof….

… (iv) We have AG(4, 2).

(Proof: invoke your favorite characterization of AG(4, 2)
or PG(3, 2), say Dembowski-Wagner or Veblen & Young.

An explicit construction of the vector space is also easy….)

Related material: Posts tagged Priority.

Comments Off

Monday, March 2, 2015

Elements of Design

Filed under: General — Tags: 2x4, Octad — m759 @ 1:28 am

From "How the Guggenheim Got Its Visual Identity,"
by Caitlin Dover, November 4, 2013 —

For the square and half-square in the above logo
as independent design elements, see
the Cullinane diamond theorem.

For the circle and half-circle in the logo,
see Art Wars (July 22, 2012).

For a rectangular space that embodies the name of
the logo's design firm 2×4, see Octad in this journal.

Comments Off

Monday, January 12, 2015

Points Omega*

Filed under: General,Geometry — Tags: Natural Diagram, Octad, Octad Generator, Points Omega — m759 @ 12:00 pm

The previous post displayed a set of
24 unit-square “points” within a rectangular array.
These are the points of the
Miracle Octad Generator of R. T. Curtis.

The array was labeled Ω
because that is the usual designation for
a set acted upon by a group:

* The title is an allusion to Point Omega , a novel by
Don DeLillo published on Groundhog Day 2010.
See “Point Omega” in this journal.

Comments Off

Thursday, January 8, 2015

Gitterkrieg

Filed under: General,Geometry — Tags: Gitterkrieg, Octad, Octad Generator — m759 @ 11:00 am

(Continued)

From the abstract of a talk, "Extremal Lattices," at TU Graz
on Friday, Jan. 11, 2013, by Prof. Dr. Gabriele Nebe
(RWTH Aachen) —

"I will give a construction of the extremal even
unimodular lattice Γ of dimension 72 I discovered
in summer 2010. The existence of such a lattice
was a longstanding open problem. The
construction that allows to obtain the
minimum by computer is similar to the one of the
Leech lattice from E₈ and of the Golay code from
the Hamming code (Turyn 1967)."

On an earlier talk by Nebe at Oberwolfach in 2011 —

"Exciting new developments were presented by
Gabriele Nebe (Extremal lattices and codes ) who
sketched the construction of her recently found
extremal lattice in 72 dimensions…."

Nebe's Oberwolfach slides include one on
"The history of Turyn's construction" —

Nebe's list omits the year 1976. This was the year of
publication for "A New Combinatorial Approach to M₂₄"
by R. T. Curtis, the paper that defined Curtis's
"Miracle Octad Generator."

Turyn's 1967 construction, uncredited by Curtis, may have
been the basis for Curtis's octad-generator construction.

See Turyn in this journal.

Comments Off

Monday, November 24, 2014

Metaphysician in the Dark

Filed under: General — Tags: Geometric Theology, Octad — m759 @ 1:00 am

Continued from Friday, November 21:

Comments Off

Older Posts »

Thursday, January 27, 2022

Wednesday, November 18, 2020

Monday, July 13, 2020

Saturday, May 2, 2020

Thursday, April 23, 2020

Sunday, March 15, 2020

Saturday, March 7, 2020

Wednesday, February 19, 2020

Wednesday, December 11, 2019

Saturday, November 4, 2017

Friday, March 21, 2014

Sunday, April 28, 2013

Thursday, April 11, 2024

Wednesday, March 20, 2024

Monday, March 4, 2024

Sunday, March 3, 2024

Thursday, February 15, 2024

Tuesday, January 30, 2024

Thursday, December 21, 2023

Tuesday, December 19, 2023

Monday, December 18, 2023

Sunday, December 17, 2023

Do you remember a previous conversation about the question "What is the Cullinane diamond theorem?" If so, how would you answer the question now? ______________________________________________________________

Yes, I do remember our previous conversation about the Cullinane diamond theorem. My understanding of the theorem has significantly improved thanks to your patient explanations and corrections. Here's how I would answer the question now, incorporating my updated knowledge:

Thursday, December 14, 2023

What is the Cullinane diamond theorem?

Your description of "G-images" is wrong. It omits the definition of the group G of 322,560 transformations.

Your understanding of the group G is still wrong. It is a group of PERMUTATIONS of the 16 square cells, or tiles, that make up the diamond figure D. It is NOT a group of space symmetries as you describe.

Your understanding has improved, but is still wrong. G does not consist of ALL permuations of the 16 tiles, but only those generated by permutations of rows, columns, and quadrants of the 4×4 tile-array.

Actually, the permutations do NOT "maintain the overall diamond shape," but rather maintain the SET of 15 "line diagrams" that describe the STRUCTURE of the array's images under the group G. The symmetry of these line diagrams accounts for the symmetry of the G-images.

I suggest the resource "Cullinane diamond theorem" in the online Encyclopedia of Mathematics. Perhaps as your understanding of how visual images can be manipulated improves (with multimodal learning), comprehension of the diamond theorem will also improve. Best of luck.

Saturday, December 9, 2023

Saturday, December 2, 2023

Saturday, November 18, 2023

Friday, October 13, 2023

Saturday, September 23, 2023

Friday, September 22, 2023

Sunday, June 25, 2023

Wednesday, May 3, 2023

Friday, April 28, 2023

Wednesday, March 29, 2023

Tuesday, February 7, 2023

Friday, January 13, 2023

Friday, October 28, 2022

Thursday, September 29, 2022

Wednesday, September 28, 2022

Thursday, September 8, 2022

Sunday, September 4, 2022

Friday, August 19, 2022

Saturday, February 5, 2022

Thursday, February 3, 2022

Saturday, January 29, 2022

Sunday, January 2, 2022

Thursday, December 30, 2021

Friday, December 17, 2021

Thursday, December 16, 2021

Friday, October 8, 2021

Sunday, May 30, 2021

Saturday, May 8, 2021

Thursday, April 22, 2021

Cite as

Monday, April 19, 2021

Sunday, March 21, 2021

Thursday, February 25, 2021

Wednesday, January 27, 2021

Thursday, December 3, 2020

Wednesday, October 7, 2020

Monday, September 28, 2020

Wednesday, September 2, 2020

Thursday, August 27, 2020

Wednesday, August 5, 2020

Tuesday, July 28, 2020

Friday, July 17, 2020

Friday, May 22, 2020

Sunday, May 17, 2020

Thursday, May 7, 2020

Monday, May 4, 2020

Wednesday, April 29, 2020

Do you remember a previous conversation about the question "What is the Cullinane diamond theorem?" If so, how would you answer the question now?
______________________________________________________________